GALOIS REPRESENTATIONS VALUED IN REDUCTIVE GROUPS AND · GALOIS REPRESENTATIONS VALUED IN REDUCTIVE...

GALOIS REPRESENTATIONS VALUED IN REDUCTIVE GROUPS AND

THEIR CENTRALIZERS

NIKOLAY GRANTCHAROV AND WYATT REEVES

Abstract. For G a classical reductive group over C, we describe representations of Gal(Qp,Qp)

modulo its wild inertia such that the image of this quotient in G has finite centralizer. For eachsuch centralizer, we also describe its representations.

1. Introduction

1.1. History. This project is motivated by a refined version of the local Langlands conjecture.Roughly speaking, the local Langlands conjecture provides a correspondence between representa-tions of a p-adic group and certain maps from the Weil group into the dual group. These maps arecalled “Langlands parameters” and are conjectured to partition the p-adic group representationsinto finite sets called “L-packets”. We explain this conjecture in its basic form below and show howit relates to our problem. For a more complete exposition of the Langlands program and the furthertechnical properties stated in the conjecture, we refer the reader for example to [Gel84],[Bor79],and [DR09].

To state the basic form of the conjecture, we need to set up some terminology. Let WQp denote

the Weil group for primes p < ∞. It is a dense subgroup of Gal(Qp/Qp) generated by inertia

subgroup I and Frobenius element Frob ∈ Gal(Fp/Fp). Let WQ/R denote the Weil group forp =∞. It is defined as the unique nonsplit extension of short exact sequence

1→ C× →WQ/R → Gal(C/R)→ 1.

Now, define the Weil-Deligne group

Lp :=

WC/R if p =∞WQp × SL2(C) if p <∞.

Then the simplified conjecture states that there is a finite-to-1 surjectionLp

Φ−→ G

←− Irr(G(Qp))

where on the left we require φ to be a “tempered, admissible Langlands parameter” [Bor79]. The

finite fibers of this map are called L-packets. The centralizer SΦ = Cent(Im(φ), G) and its rep-resentations roughly parameterize these L-packets, thus giving rise to a “refined” local Langlandscorrespondence [Kal15]. Furthermore, we are interested in the trace and dimension of these rep-resentations due to the conjectural Langlands-Shelstad transfer factors [LS87]. Numerous effortshave been made in studying and constructing these L-packets, see for example [Kal16],[DR09],and [Yu01]. In particular, Kaletha reduced a refined version of the local conjecture to the case of“depth-zero Langlands parameters”, i.e when Φ is trivial on SL2(C) and wild inertia. Our projectis to carefully study the depth-zero supercuspidal Langlands parameters.

Date: September 12, 2018.

1

2 NIKOLAY GRANTCHAROV AND WYATT REEVES

1.2. Set-up of the problem. In what follows, let G denote a connected, reductive group withcoefficients in C. Let Gder denote its derived subgroup, and let Gsc, Gad denote the simply connectedcover and adjoint quotient of Gder, respectively. It is a fact that Gad is in 1-1 correspondence withirreducible root systems, there are the classical types An, Bn, Cn, Dn, and the exceptional typesE6, E7, E8, F4, G2. In this paper, we focus on the classical types.

Let Fp be the finite field of p elements and let Γp := Gal(Qp/Qp). Let φ denote the Frobenius

element φ : x→ xp, which is a (topological) generator of Gal(Fp/Fp) = Z, the profinite completion

of the integers. Define the inertia subgroup I = ker

(Γp → Gal(Fp/Fp)

). Then the wild inertia

I+ C I is the maximal pro-p subgroup, and the tame inertia group is I/I+. The tame inertiais (noncanonically) isomorphic to the direct product over all primes q 6= p of the rings of q-adic

integers [Ser71, Section 4], hence is denoted by Z(p). We obtain a split short exact sequence

1→ Z(p) → I+\Γp → Z → 1.

Hence I+\Γp ∼= 〈s〉 o 〈φ〉 for some generator s ∈ Z(p) with relation φsφ−1 = sq, where q is somepower of p. We will pursue two main goals in this paper.

(1) List all representations Φ : I+\Γp → G such that SΦ := Cent(Im(Φ), G) is a finite group,(2) For each such Φ, list all irreducible representations ρ : SΦ → GLn(V ) and their character

values.

Observe, the representation Φ defined here is related to the depth-zero Langlands parameter Φ :Lp → G defined in Subsection 1.1 above.

1.3. Main Results. To achieve goal (1), it is enough to specify the image of (s, φ) so that

• s ∈ G is “regular”, “semi-simple”, and of finite order coprime to p

• φ ∈ G normalizes the torus T , T φ is finite, and• φsφ−1 = sq.

Let w0 denote the projection of φ ∈ N(T ) to w0 ∈ W , the Weyl group. We now describe some

elements of the Weyl group acting on the cocharacter lattice X∗(T ). This will help present a“normal form” of w0, i.e a nice presentation of w0 in which w0 is in general conjugate to. For a

classical root system of rank n, define elements of W with action on X∗(T ) by ei : εi → −εi; ande′i : εi → −εn+1−i, εn+1−i → εi; and “little mirror” m : ε∗i → ε∗n+1−i .

For type Bn and Cn, define w′0 ∈ W by w′0 = (1, . . . , i1) . . . (im−1 + 1, . . . , im), i0 := 0 < i1 <i2 · · · < im. This notation means w′0 : ε∗j → ε∗j+1for j /∈ i1, . . . , im and w′0 : ε∗ij → −ε

∗ij−1+1.

For Dn, due to simplifications of computation later on, define a slight variant of w′0. Namely, let

v′0 = (i1, . . . i2 − 1) . . . (im, . . . , im+1 − 1).

Let s = (s1, . . . , sn) be a representative of an element in T ∼= (C ⊗Z X∗(T ))/X∗(T ) in the ε∗icoordinates. Table (1.3.1) presents the pairs of (s, φ) which satisfy the first two conditions, whereφ is some lift of a w0 in normal form. Note that in the table we only specified the action of φ onthe first ‘half’ of coordinates of s in types Bn, Cn, Dn; the action on the second half is just a “bigmirror” of the action. Also, if we further require φsφ−1 = sq, we direct the reader to item (3) ofSection 4, to see what additional restrictions are placed.

To achieve goal (2), we use the short exact sequence

(1.3.1) 1→ T φad → Cent(s, φ; Gad)→W φs → 1

and Clifford theory to classify representations of Cent(s, φ, Gad). We say the extension Cent(s, φ, Gad)in (1.3.1) is split if the short exact sequence has a splitting. We say it is virtually split if its ir-reducible representations arise in the same way as if it were split. Section 3.4 provides a precisecharacterization of an extension being virtually split.

GALOIS REPRESENTATIONS VALUED IN REDUCTIVE GROUPS AND THEIR CENTRALIZERS 3

Table 1.3.1. Results for Part I

Type s ∈ Tad w0 ∈WAn (e2πitn+1 , . . . , e2πit1) w0 = (1, 2, . . . , n+ 1)i

d/(n+ 1) > td > · · · > t1 = 0, td+i = d/(n+ 1) + ti for some i.

Bn (e2πit1 , · · · , e2πitn , 1, e−2πitn , · · · , e−2πit1) w0 = w′0t1 = 1/2 > t2 > · · · tn > 0 i1 = 1, im = 2n

Cn (e2πit1 , . . . , e2πitn , e−2πitn , . . . , e−2πit1) w0 = ew′0mw′0m−1

ti + tn+1−i = 1/2 and ti > ti+1 im = bn/2c. e = id if n even

e : ε(n+1)/2 → −ε(n+1)/2 if n odd

D2n (e2πit1 , · · · , e2πit2n , e−2πit2n , · · · , e−2πit1) w0 = v′0mv′0m−1

ti + t2n−i+1 = 1/2, t1 = 1/2 > t2 > · · · > t2n−1 > 0 i1 = 1, i2 = 2, im < im+1 := n+ 1

D2n+1 (e2πit1 , · · · , e2πit2n , 1, e−2πit2n , · · · , e−2πit1) w0 = e′1en+1v′0mv

′−10 m−1

ti + t2n−i+2 = 1/2, t1 = 1/2 > t2 > · · · > t2n > 0 i1 = 2, im < im+1 := n+ 1

We will also be interested in the representation theory for T+, C+, which are the preimages of

Tad,Cent(s, φ, Gad), respectively, into the simply-connected cover. Table (1.3.2) fully records the

representation theory for both Cent(s, φ, Gad) and C+.

Table 1.3.2. Results for Part II

Type T φad T+ Ws Cent(s, φ; G) C+

A2n Z/(2n+ 1)Z (Z/(2n+ 1)Z)2 Z/(2n+ 1)Z split split

A2n+1 Z/(2n+ 2)Z (Z/(n+ 1)Z)× Z/(4n+ 4)Z Z/(2n+ 2)Z split virtually-split

Bn (Z/2Z)m (Z/4Z)× (Z/2Z)m−1 Z/2Z split virtually-split

Cn (Z/2Z)m or (Z/2Z)m+1 or Z/2Z split virtually-split

(Z/2Z)m−2 × Z/4Z (Z/2Z)m−1 × Z/4Z

D2n (Z/2Z)m or (Z/2Z)m−2 × Z/4Z Z/2Z× Z/2Z split virtually-split

(Z/2Z)m−2 × Z/4Z

D2n+1 (Z/2Z)m−2 × Z/4Z (Z/2Z)m−1 × Z/8Z Z/4Z virtually split virtually split

1.4. Acknowledgements. We would like to graciously thank our advisor Dr. Tasho Kaletha. Hegenerously devoted his time, throughout the extent of the program, in explaining the appropriateconcepts and background needed for our project in a very clear and concise way. We would alsolike to thank the organizers of the University of Michigan REU for making this project possible.

2. Preliminaries

2.1. Notation. Let T ⊂ B be a maximal torus and Borel subgroup of the reductive group G and

let Φ be the corresponding root system of T in G, with positive system Φ+ corresponding to B.

Let ∆ = α1, . . . , αn be the corresponding simple roots, where n is the rank of G.

Let X∗(T ) be the character lattice of algebraic homomorphisms from T to C×, and X∗(T ) be the

cocharacter lattice of algebraic homomorphisms from C× to T . Composition defines a Z-bilinearpairing

〈 , 〉 : X∗(T )×X∗(T )→ Hom(C×,C×) = Z.


Since T ∼= (C×)n, we know that both X∗(T ) and X∗(T ) are free abelian groups of rank n and thatthe pairing is nondegenerate.

Now define two vector spaces

V = C⊗Z X∗(T )

and

V ∗ = C⊗Z X∗(T ).

These are both n-dimensional C-vector spaces, and we can extend 〈 , 〉 to a nondegenerate C-bilinear pairing V ∗ × V → C. Having done this, define the coweights ω1, . . . , ωn in V such that

ωi is dual to αi with respect to the pairing. Furthermore, we can naturally identify X∗(T ) and

X∗(T ) with the subgroups Z⊗ZX∗(T ) of V ∗ and Z⊗ZX∗(T ) of V , respectively. We conclude that

X∗(T ) and X∗(T ) are dual to each other with respect to the pairing.Given a root α ∈ Φ, there is a unique element α ∈ V such that 〈α, α〉 = 2 and the map

σα : V ∗ → V ∗ x 7→ x− 〈x, α〉α

preserves Φ. Collectively, these elements are called coroots, and are denoted Φ.

For a general reductive group G, we have the relation

Q ⊂ X∗(T ) ⊂ P

where P is the coweight lattice, i.e the Z-span of coweights ω, and Q is the coroot lattice = Z(Φ).

Moreover it is known P = X∗(T ) if and only if G is adjoint and X∗(T ) = Q if and only if G issimply-connected.

Let N(T ) denote the normalizer of T in G and W := N(T )/T denote the Weyl group. Note that

W acts on T by conjugation, making T into a W -module. This defines a W -module structure on

X∗(T ) which can be extended to one on C ⊗Z X∗(T ) by letting W act trivially on C. With thisaction, V becomes a C-linear W representation.

With these W -module structures, the Z-bilinear map

C×X∗(T )→ T (t, ω) 7→ ω(e2πit

)gives rise to a W -module homomorphism

exp : V → T .

This map is surjective, and its kernel is Z⊗ZX∗(T ) ∼= X∗(T ), so we obtain the following isomorphismof W -modules

T ∼= V/X∗(T ).

Finally, note that for any x ∈ V and any α ∈ X∗(T ), we have

(2.1.1) α(exp(x)) = e2πi〈α,x〉.

2.2. Inner regular elements. In this section, we will be concerned with pairs (s, ϕ) of automor-

phisms of Gad which are in particular inner - i.e the s, ϕ action on Gad is given by conjugation by

s, ϕ, respectively. So Gsad is understood as the Ad-s invariants of G. We will see that on the Liealgebra level, an inner element is conjugate to an element of the standard alcove. Our followingexposition closely follows that of [R09].

An element s ∈ Gad is semisimple if s acts diagonally on g := Lie(Gad). Any torsion element

s ∈ Gad is semisimple and Gad-conjugate to an element of Tad, and there is x ∈ VQ := Q⊗ P such

that s = exp(x). We have x, x′ ∈ VQ give rise to G-conjugate elements exp(x), exp(x′) if and onlyif x, x′ are conjugate under the extended affine Weyl group

W := W n P .


Here, P acts on V by translations. The (unextended) affine Weyl group is the normal subgroup

W := W n QC W .

It can also be thought of as the group of reflections in V about the affine root hyperplanes α = nfor α ∈ Φ and n ∈ Z. We call C an alcove if it is a connected component of the set of x ∈ V notlying in any root hyperplane. A wall of C is the intersection of a root hyperplane Hα,n with the

alcove closure C. Thus each alcove has n + 1 walls. From [Bou02, V.3.2], W is a Coxeter group

generated by the n+ 1 hyperplanes about a fixed alcove and W permutes the alcoves in V freelyand transitively. We now describe the standard alcove that will be used for our computations.

To the base ∆ = α1, .., αn, let α =∑n

1 aiαi be the highest root. Let α0 := 1− α be an affinelinear function, so

n∑i=0

aiαi = 1,

where a0 := 1. Then

C = x ∈ V : 〈αi, x〉 > 0 for 0 ≤ i ≤ nis the standard alcove associated to base ∆. In terms of barycentric coordinates,

C = n∑i=0

xiωi/ai : xi > 0 andn∑i=0

xi = 1,

where ω0 := 0. The following information can be also found in [Bou02, VI]. Since W acts transi-

tively on the set of alcoves, W does too. Hence C intersects all W -orbits in V . This means each

torsion element s ∈ Gad is conjugate to exp(x) for some x ∈ C ∩ VQ. However, the extended affine

Weyl group W does not act freely on the alcoves in V like W , so we consider the alcove stabilizer

Ω := ρ ∈ W : ρ · C = C.

We have the decomposition

W = Ω n W .

It then follows for x, x′ ∈ C, exp(x) and exp(x′) are Gad-conjugate if and only if x and x′ areconjugate under Ω.

It is useful to construct an explicit isomorphism

Ω ∼= X∗(T )/ZΦ.

For each coset in X∗(T )/ZΦ, there exists a unique coweight ω which is a vertex of alcove C. Wecall such a coweight miniscule. For each miniscule coweight, there is exists a unique ρi ∈ Ω suchthat ρi · ω0 = ωi. This bijection is a group homomorphism.

We call an element s ∈ G regular if the identity component of its centralizer Cent(s; G) is a

maximal torus. We call an element s ∈ T strongly regular if Cent(s; G) is a maximal torus. By[R09, Prop 2.1],

Cent(s; G)/Cent(s; G) ∼= Ωx := ρ ∈ Ω : ρ · x = x,so s = exp(x) is strongly regular precisely when x is not fixed by an element of the affine Weylgroup that stabilizes the alcove. Furthermore, Ωx

∼= Ws/Ws , where Ws is the stabilizer of s in W

and W s is generated by reflections for the roots in α ∈ Φ+ : α(s) = 1. To visualize the action

of Ω on x =∑n

i=0 siωi, label the nodes of the extended Dynkin diagram D(g) by si; Ω acts on x

via the symmetries of D(g). This gives an efficient way of explicitly describing the regular, but notstrongly regular elements.


Consider an element s ∈ G which is semisimple, regular, but not strongly regular. Then W s is

trivial, Ωx = Ws, Cent(s; G) = T , and we have the short exact sequence

0→ T → Cent(s; G)→Ws → 0

Let ϕ ∈ N(T ) be such that Tϕ is finite and ϕsϕ−1 = sq for q a power of a prime. Denote by

w0 the projection of ϕ ∈ N(T ) onto the Weyl group W . Then Cent(s; G)ϕ = Cent(s, ϕ; G) and wehave the exact sequence

0→ Tϕ → Cent(s, ϕ; G)→Wϕs → H1(〈ϕ〉, T )→ · · ·

Observe

0→ Tϕ → Tϕ−1−−→ T

is an exact sequence of algebras, and since Tϕ = ker(ϕ− 1) is finite, ϕ− 1 is surjective. Hence the

first cohomology group H1(〈ϕ〉, T ) = ker(N)/im(ϕ − 1) = T /T = 1, where N =∑ord(ϕ)−1

0 ϕi isthe norm map, and we have proven

0→ Tϕad → Cent(s, ϕ; Gad)→Wϕs → 1

is an exact sequence. Furthermore, if one takes T+ and Cent(s, ϕ, Gad)+ to be the preimages of Tϕad

and Cent(s, ϕ; Gad) in Gsc, we have the commutative diagram

(2.2.1)

1 T+ Cent(s, ϕ, Gad)+ Wϕs 1

1 Tϕad Cent(s, ϕ, Gad) Wϕs 1

||

We conclude this subsection with a useful criterion for the group of toral ϕ-fixed points to befinite.

Lemma 2.2.1. For any ϕ ∈ N(T ), Tϕ is finite if and only if the linear operator ϕ − 1 : V → V

is invertible. If this is the case, then |Tϕ| = det(ϕ − 1) = pϕ(1), where pϕ is the characteristicpolynomial of ϕ.

Proof. Explicitly, Tϕ = t ∈ T | (ϕ · t)t−1 = 1. Since exp : V → T is a surjective group

homomorphism, Tϕ ∼= Fϕ/X∗(T ), where Fϕ is the preimage of Tϕ under exp. Because exp is

ϕ-equivariant, we can write Fϕ explicitly asx ∈ V | ϕ · x− x ∈ X∗(T )

. Now consider the linear

operator ϕ − 1 acting on V . If for some x ∈ V we know x ∈ ker(ϕ − 1), then x is surely in

Fϕ. If Fϕ/X∗(T ) is finite, Fϕ must be covered by a finite number of translates of X∗(T ). Sinceker(ϕ − 1) ⊆ Fϕ, we know ker(ϕ − 1) is covered by those same translates. We also know thatker(ϕ − 1) ∼= Cn for some n, so n = 0. Since ϕ − 1 is injective and sends V → V , it must be anisomorphism.

On the other hand, suppose that ϕ − 1 is invertible. Then we can describe Fϕ explicitly as

(ϕ − 1)−1X∗(T ), so Fϕ is also a free abelian group of rank n. Since ϕ maps X∗(T ) into itself, we

see that (ϕ− 1) is an injective endomorphism of Fϕ with image X∗(T ). As a result, Fϕ/X∗(T ) is

finite. Furthermore, using the Smith Normal Form for ϕ − 1, we see that |Tϕ| = |Fϕ/X∗(T )| =det(ϕ− 1) = pϕ(1).

Corollary 2.2.2. Whether Tϕ is finite depends only on the conjugacy class of ϕ in W .

Corollary 2.2.3. Whether Tϕ is finite doesn’t depend on the choice of group in a given isogenyclass.


2.3. Basic properties of G. The tables below provide fundamental properties of G. The root

system provided is for Lie(G). Here, ∆ = α1, .., αn, ∆ = α1, .., αn, and α is the highest rootwith respect to ∆.

Table 2.3.1. Root Systems

Type Φ ∆, ∆ α

An εi − εj : i 6= j ε1 − ε2, .., εn − εn+1 α1 + ..+ αn

ε∗1 − ε∗2, .., ε∗n − ε∗n+1Bn ±εi,±(εi ± εj) : i 6= j ε1 − ε2, .., εn−1 − εn, εn α1 + 2α2..+ 2αn

ε∗1 − ε∗2, .., ε∗n−1 − ε∗n, 2ε∗nCn 2εi,±(εi ± εj) : i 6= j ε1 − ε2, .., εn−1 − εn, 2εn 2α1 + 2α2..+ 2αn−1 + αn

ε∗1 − ε∗2, .., ε∗n−1 − ε∗n, ε∗nDn ±(εi ± εj) : i 6= j ε1 − ε2, .., εn−1 − εn, εn−1 + εn α1 + 2α2 + ..+ 2αn−2 + αn−1 + αn

ε∗1 − ε∗2, .., ε∗n−1 − ε∗n, ε∗n−1 + ε∗n

Table 2.3.2. Weyl Groups

Gad Gsc W Ω

PSLn+1 = SLn+1/µn+1 SLn+1 Σn+1 Z/(n+ 1)Z

SO2n+1 Spin2n+1 (Z/2Z)n o Σn Z/2Z

PSp2n = Sp2n/µ2 Sp2n (Z/2Z)n o Σn Z/2Z

PSO2n = SO2n/µ2 Spin2n (Z/2Z)n−1 o Σn n even: (Z/2Z)2

n odd: Z/4Z

3. Representation theory for Group Extensions

3.1. Group Extensions. In this section we state the theory needed to describe irreducible repre-

sentations of Cent(s, ϕ, Gad) and Cent(s, ϕ, Gad)+, both of which are extensions of a finite abeliangroup by a finite abelian group.

First let us recall some basic results of group extensions and Clifford theory. Given a group G,and an abelian group K, we say E is an extension of K by G if it fits in the short exact sequence

1→ K → E → G→ 1

Choose a set-theoretic section s : G→ E. This defines a G action on K by

g · k := s(g)ks(g)−1.

Also, s determines a 2-cocyle

ξs(g1, g2) := s(g1)s(g2)s(g1g2)−1 ∈ Z2(G,K),

which “measures” how far s is from being a group homomorphism. Changing the section s replacesξ by a cohomologous 2-cocycle, so the class [ξ] ∈ H2(G,E) is independent of choice of section.

Given a group G, a G-module K, and a normalized 2-cocycle ξ ∈ Z2(G,K), we can constructthe group K ξ G whose underlying set is the Cartesian product K ×G, and has the group law

(k1, g1)(k2, g2) :=(k1g1k2ξ(g1, g2), g1g2

)


(here g1k2 = g1 · k2). Now if we have an extension E of K by G, with a chosen section s, we seethere is an isomorphism E → K ξs G given by sending e = ks(g) ∈ E to (k, g).

A short lemma now shows the section determines the group structure on E completely.

Lemma 3.1.1. [Bro82, Section 4.3] There is a 1− 1 correspondenceEquivalence classes of extensions of K by G

with prescribed G-action on K

←→ H2(G,K)

given by

[E] −→ [ξs]

K ξ G←− [ξ]

where the “prescribed action” is the one determined by a choice of section. E and E′ are said tobe equivalent extensions if there exists a commutative diagram of short exact sequences.

3.2. Clifford theory. Clifford theory is the study of representations of a finite group extension Gin terms of that of its normal subgroup N and quotient H. We will now quote two lemmas from[Kal18] on the subject. In what follows, Irr(G) will denote the isomorphism classes of irreducibleG-representations.

Lemma 3.2.1. Suppose we have an exact sequence of finite, not necessarily abelian groups

1→ N → G→ H → 1

Let π ∈ Irr(G) and Sπ ⊂ Irr(N) denote the irreducible N -representation occurring in π|N . Then

(1) Sπ is a single H orbit and each member of Sπ occurs with the same multiplicity, denotedmπ.

(2) The map

Irr(G)→ Irr(N)/H

defined by π → Sπ is surjective.

Lemma 3.2.2. Suppose H is abelian and σ ∈ Irr(N). Let Gσ := Stab(σ,G) and G′σ ⊂ Gσ be thelargest subgroup to which σ extends, i.e σ is a (linear) representation of G′σ such that σ|N = σ.Then

IndGG′σ σ

is an irreducible G-representation and every irreducible G-representation arises this way.

It is easy to show that when Hσ = Stab(σ,H) is cyclic, then G′σ = Gσ. Using the previouslemma, we can recover the classical classification of irreducible representations for a semi-directproduct of abelian normal group by quotient:

Corollary 3.2.3. [Ser71, Section 8.2] Suppose G = N oH for an abelian normal subgroup N andsubgroup H of G. Let χ : N → C× be a character and let

Hχ = h ∈ H : χ(hnh−1) = χ(n), ∀n ∈ N.Then extend χ to the character χ : N o Hχ → C× by setting χ(nh) = χ(n). For any irreduciblerepresentation ρ of Hχ, composing with the standard projection N oHχ → Hχ gives an irreduciblerepresentation ρ′ of Gχ := N oHχ. Then

IndGGχ(χ⊗ ρ′)

is an irreducible G-representation and every irreducible G-representation is uniquely determinedthis way.

We now want to generalize this result to the case when the extension is not split.


Lemma 3.2.4. Suppose we have the exact sequence of groups 1 → A → B → C → 1 where A isabelian, and a given representation χ of A. Choose a set-theoretic section s : C → B and identifyB ∼= A s C. Then χ is an extension of χ to B if and only if χ(a, c) = χ(a)σ(c), where σ is aset-theoretic function C → C× that satisfies χ ξs = ξσ.

Proof. Given an extension χ of χ to B, define σ : C → C× to be the function c 7→ χ(1, c). Thenbecause χ extends χ, we have that

χ(a, c) = χ(a, 1)χ(1, c) = χ(a)σ(c).

Also, since χ is a group homomorphism B → C×, we know that

1 = χ(a1, c1)χ(a2, c2)χ(a1c1a2ξs(c1, c2), c1c2)−1

= χ(a1)σ(c1)χ(a2)σ(c2)σ(c1c2)−1χ(a1c1a2ξs(c1, c2))−1.

Since C× is abelian and χ is a homomorphism, this implies that

(3.2.1) 1 = σ(c1)σ(c2)σ(c1c2)−1χ(a2)χ(c1a2)−1χ(ξs(c1, c2)))−1,

but because (c1a2, 1) = (1, c1)(a2, 1)(1, c1)−1, we see that

χ(c1a2) = χ((1, c1)(a2, 1)(1, c1)−1

)= χ(a2, 1) = χ(a2),

so equation 3.2.1 becomes

(3.2.2) χ ξs(c1, c2) = ξσ(c1, c2).

On the other hand, if we choose a σ : C → C× that satisfies equation 3.2.2 and define thefunction χ : B → C× by χ(a, c) = χ(a)σ(c), then the calculation we just did shows that χ is agroup homomorphism. Also,

σ(1) = σ(1)σ(1)σ(1)−1 = ξσ(1, 1) = χ ξs(1, 1) = 1,

so χ extends χ.

With the notation being the same as in lemma 3.2.4, we make the following definition:

Definition 3.2.5. Suppose 1 → A → B → C → 1 is an exact sequence where A and C areabelian. Then, with notation as in 3.2.3, for a given character χ of A, we obtain an exact sequence1 → A → Bχ → Cχ → 1. If there exists a set-theoretic section s : Cχ → Bχ such that χ ξs = 1,then we call χ a virtually split character. If all of the characters of A are virtually split in this way,then we say that B is a virtually split extension of A by C.

All split extensions are virtually split. When B is a virtually split extension, by lemma 3.2.4 wecan extend any character χ of A trivially to χ by setting χ(as(c)) = χ(a). Then by lemma 3.2.2all irreducible representations of B are given by

IndBB′χχ⊗ σ

for characters χ of A and σ of C (here we regard σ as a character of B by pulling it back along themap B → C. In particular, the irreducible representations of virtually split extensions arise in theexact same way as irreducible representations of split extensions (by lemma 3.2.2).

3.3. Langlands-Shelstad section. It is a result of Langlands and Shelstad [LS87] that for a

connected reductive group G, a pinning (T , B, Xα) of G gives a section n : W → N(T ) with aparticularly nice 2-cocycle formula. The construction is as follows:

For a simple root α, write Hα for the image of α under the identification of V with Lie(T ). Next,

choose X−α ∈ Lie(G) to be the unique element such that [Xα, X−α] = Hα. The subspace of Lie(G)

spanned by Xα, Hα, X−α is a Lie subalgebra of Lie(G) which is isomorphic to sl2(C). Because G


is connected, there is a unique homomorphism of algebraic groups ϕ : SL2(C)→ G whose tangentmap at the identity is this inclusion sl2(C) → Lie(G). Furthermore,

ϕ

[0 −1

1 0

]acts on T in the same way as σα ∈ W (T ), so define n(σα) to be equal to this element of N(T ).Then for a typical element σ ∈ W , express σ as a reduced composition of reflections about simpleroots σ = σα1 . . . σαn and define n(σ) = n(σα1) . . . n(σαn). This is well defined, because n(σ) isindependent of the choice of reduced expression for σ. Finally, set n(1) = 1.

Lemma 3.3.1. [LS87, Lemma 2.1A] The 2-cocycle ξ(σ1, σ2) = n(σ1)n(σ2)n(σ1σ2)−1 ∈ Z2(W, T )is given by the formula

ξ(σ1, σ2) =∏α>0

σ−11 α<0

σ−12 σ−1

1 α>0

α(−1),

where α > 0 means α is a root in B and α ∈ X∗(T ) is the coroot associated to α.

3.4. Concrete Conditions for Virtual Splitness. Given some ϕ ∈ N(Tad) with |Tϕ| <∞, we

can project ϕ to some w0 ∈ W and lift w0 to N(Tsc). Since Tϕ is finite, any lift of w0 to N(Tsc)

can be conjugated to any other by an element of Tsc. Conjugating N(Tsc) by an element of Tsc

replaces the extension to an isomorphic one. As a result, if we have some section n : W → Nsc and

we write p for the projection Gsc → Gad we may without loss of generality lift w0 to n(w0) andreplace ϕ with p n(w0).

We now describe a procedure for lifting an element from Wϕ to Cent(ϕ; Gad)+ and provideconditions for this lift to give a splitting or “virtual splitting”. For commuting w, v ∈W , let

c(w, v) := n(w)n(v)n(w)−1n(v)−1 = ξ(w, v)ξ(v, w)−1

denote the commutator. Now choose a lift c(w,w0) of c to V such that expsc c = c. Since ϕ − 1acts invertibly on V by (2.2.1), we can form

(3.4.1) tw = (ϕ− 1)−1c(w,w0) ∈ V.Define tw := expad(tw). If we apply expad to the equation (ϕ − 1)tw = c(w,w0) and keep in mindexpad = p expsc, we see that ϕtw = p c(w,w0)tw.

Now define the section

s : Wϕ → N(Tad) w 7→ tw(p n(w)),

and note that

ϕs(w) = ϕtwϕ(p n(w))

= tw(p c(w,w0))(p c(w0, w))(p n(w))

= tw(p n(w))

= s(w),

so s actually maps Wϕ into Cent(ϕ, Gad).

In what follows, we will use the convention s for a section from Weyl group to centralizer, andn for section from Weyl group into normalizer. If we now define t′w = expsc(tw) and define

s′ : Wϕ → Gsc w 7→ t′wn(w),

we see that p s′(w) = s(w), so that s′ actually maps into Cent(ϕ;Gad)+.


If we now choose a lift ξ of ξn to V along expsc, for any two w, v ∈Wϕ, we can define

A(w, v) := c(w,w0) + w · c(v, w0)− c(wv,w0) + (ϕ− 1)ξ(w, v)

and define B(w, v) := (ϕ− 1)−1A(w, v). Note that

ξs(w, v) = s(w)s(v)s(wv)−1

= tw(p n(w))tv(p n(v))(p n(wv))−1t−1wv

= twwtv(p ξn(w, v))t−1wv

= expad(B(w, v)),

and similarly, ξs′ = expsc B. As a result, if B(w, v) ∈ P for all w, v ∈W φs , we see that ξs is trivial

on W φs , so s gives a splitting of the exact sequence 1→ Tϕ → Cent(s, ϕ; Gad)→Wϕ

s → 1.Now suppose that

(3.4.2) A(w, v) = (ϕ− 1) · q + (w − 1) · p′

for some p′ ∈ P and q ∈ Q. Define f := (ϕ − 1)−1p′ and note that since (ϕ − 1)f = p′, we have

expad(f) ∈ Tϕad and therefore expsc(f) ∈ T+. If we act on (3.4.2) by (ϕ − 1)−1 and note that

(ϕ− 1)−1 and (w − 1) commute, we see that (w − 1)f = B(w, v)− q.Now let χ be a character of T+ and let w and v be elements of (Wϕ

s )χ, the subgroup of W φs

which stabilizes χ. If we define f ′ = expsc(f), we find

χ(ξs′(w, v)) = χ(expsc(B(w, v)))

= χ(expsc((w − 1)f))

= χ(wff−1)

= χ(wf)χ(f−1)

= χ(f)χ(f)−1

= 1,

so that χ is virtually split. Since if (3.4.2) holds, this is true for all characters of T+, we see that in

this case Cent(s, φ;Gad)+ is virtually split. A similar argument using ξs and f = expad(f) showsthat if (3.4.2) holds, then Cent(s, φ;Gad) is virtually split.


4. Case Study

We will systematically perform the following computations using the theory developed fromsections 2 and 3:

(1) Describe the regular, but not strongly regular elements s ∈ Gad.

(2) Describe the ϕ ∈ N(Gad) such that Tϕad is finite and Wϕs is non-trivial.

(3) Describe pairs (s, ϕ) satisfying (1), (2) and ϕsϕ−1 = sq for q a power of a prime.

(4) Describe Cent(s, ϕ; Gad) for such pairs (s, ϕ) and its irreducible representations.

(5) Describe Cent(s, ϕ; Gad)+ for such pairs (s, ϕ) and its irreducible representations.

For root systems of the classical type, we will begin by introducing a convenient coordinate systemto simplify the computation.

4.1. Type An. Since there is an isomorphism PGLn(C) ∼= PSLn(C), we will work with PGLn(C)for notational convenience.

(1) Choose the standard maximal torus of diagonal matrices and the standard Borel subgroupof upper-triangular matrices. In this case, αi ∈ ∆ is the character αi(t) = ti/ti+1.

In type An, we know the highest root is α1 + . . .+αn, so vi = ωi for all i. The extended Dynkindiagram for type An is

The alcove stabilizer Ω acts on D(g) via rotations, so Ω ∼= Z/(n+1)Z. Write w for the generatorof Ω which sends vi 7→ vi+1. The semisimple elements of V fixed by a nontrivial rotation wd ∈ Ωare given by

x =n∑i=0

sivi

such that si = si+d for all i = 0, 1, .., n, and∑n

0 si = 1 and si > 0 for all i. Then by Equation 2.1.1and Table 2.3.1,

(4.1.1) (e2πitn+1 , . . . , e2πit1) ∈ PGLn+1(C),

where and the ti satisfy d/(n+ 1) > td > · · · > t1 = 0 and td+i = d/(n+ 1) + ti for all i.

(2) Since ϕ acts on the space W spanned by ε∗1 . . . ε∗n as a permutation matrix, its characteristic

polynomial is pϕ,W (λ) =∏i λ

ì − 1, where ì is the length of the ith cycle of ϕ. Since this actiondecomposes into an action on V and an action on C(ε∗1 + . . . + ε∗n) (the second of which ϕ acts

on trivially) we see that pϕ,W (λ) = pϕ,V (λ)(λ − 1). If Tϕ is finite, then pϕ,V (1) 6= 0, so pϕ,W (λ)

can contain only a single factor of λì − 1, so w0 must be a single cycle in Σn+1. In this case,

pϕ,V (λ) = λn + λn−1 + . . . + 1, so |Tϕ| = pϕ,V (1) = n + 1. If necessary, conjugate w0 so that it is

the cycle (1 2 . . . n+1) ∈ Σn+1. Having done this, we can write the elements of Tϕ explicitly: write

ζ = e2πi/(n+1). Then the ith element of Tϕ is ti = [diag(1, ζi, . . . , ζi(n−1))]. From this, we see that

Tϕad∼= Z/(n+ 1)Z

via the isomorphism ti 7→ i.If Wϕ

s 6= 1, there exists w′ 6= 1 such that w0 commutes with it and

Wϕs = 〈w′〉 ∼= Z/n+1

dd′ Z,

where w′ = wdd′, d′|(n + 1)/d. Lastly, since w0 is a cycle which commutes with w′, there exists i

such that wi0 = w′ and gcd(i, n+1) = dd′ For simplicity, we assume d′ = 1, w′ = wd in the followingsteps.


(3) We wish to solve the equation

w0.(sn+1, . . . , s1) = (spn+1, . . . , sp1) ∈ PGLn+1(C)

for s = (sn+1, . . . s1) as given in (4.1.1). This amounts to solving s1 = spw0(1)z, . . . , swi−10 (1)z = spd+1z

where z ∈ C×. Since our solution is in PGLn+1(C), we may as well take s1 = 1. Since w0 is acycle, all the si are now determined once we specify z. After composing the equations i times, we

have sj = spi

j+dzpi−1p−1 , and using the fact that sj+d = ζdsj , we see that

sj = spi

j zpi−1p−1 ζp

id

for all j = 1, . . . n + 1. In particular for j = 1, we have sj = 1 which implies (†) zpi−1p−1 = ζ−p

id.Observe

ζd = sd+1 = swi0(1)

= spwi+1

0 (1)z

= spd+w0(1)z

= ζdpspw0(1)z = ζdp,

which implies d(1− p) ≡ 0 mod n+ 1, or equivalently, p ≡ 1 mod (n+ 1)/d. So we can simplify

(†) to get zpi−1p−1 = ζ−d, so z is some pi−1

p−1 root of ζ−d.

Now, it just remains to check for a given z whether we have the string of inequalities arg(sn+1) >arg(sn) > · · · > arg(s1) = 0. We see our work fully classify triples (s, φ, p) satisfying steps(1), (2), (3). Furthermore once we specify p, we see there are only finitely many pairs (s, φ), whichcan all be found systematically using a computer program.

(4) The map n : W → N(T ) which maps every element of W to the equivalence class of its

permutation matrix, w 7→ [P (w)], is a splitting of W into N(T ). Since Tϕad is finite, by conjugating

ϕ by an element of Tad we can fix ϕ = n(w0). Then the restriction of n to Wϕs gives a splitting of

Wϕs into Cent(s, ϕ; Gad). Since each element w ∈Wϕ

s is a power of w0, it acts trivially on Tϕad. Asa result,

Cent(s, ϕ; Gad) ∼= Z/(n+ 1)Z× Z/n+1k Z.

(5) We now want to understand the pullback C+ of Cent(s, ϕ; Gad) to Gsc = SLn+1(C). First,note that

det(diag(1, ζi, . . . , ζi(n−1)) =

n−1∏j=0

ζij =

n∏j=1

ζij = ζin(n+1)/2.

Also note that

det(P (ϕ)) = sign(1 2 . . . n+ 1) = (−1)n.

Case 1: n+ 1 odd.In this case, ζin(n+1)/2 = 1, so the map r : Tϕad → T+ sending ti 7→ diag(1, ζi, . . . , ζi(n−1)) gives a

splitting of Tϕ into T+. Since T+ is abelian, we conclude

T+∼= (Z/(n+ 1)Z)2.

Also, since det(P (wi0)) = det(P (w0))i = 1, the map s : Wϕs → C+ sending w 7→ P (w) gives a

splitting of Wϕs into C+, so that

C+∼= (Z/(n+ 1)Z)2 o Z/n+1

k Z.


Explicitly, since w−10 · r(ti) = ζir(ti), we see that an element w = w−`0 ∈W

ϕs acts on T+ by sending

ζjr(ti) to w · ζjr(ti) = ζj+ir(ti).

Case 2: n+ 1 even.In this case, ζin(n+1)/2 = (−1)i. As a result, the map r : Tϕad → T+ which sends

ti 7→

diag(1, ζi, . . . , ζi(n−1)) 2|iζ1/2diag(1, ζi, . . . , ζi(n−1)) 2 - i

gives a set-theoretic section Tϕad → T+. Its cocycle is

ξr(ti, tj) =

ζ 2 - i and 2 - j1 otherwise.

Because of this, we see r(t1)i = ζbi/2cr(ti), so r(t1) has order 2(n + 1). On the other handr(t2)i = r(t2i), so r(t2) has order (n+ 1)/2. Since r(t2)i always has a 1 in its first coordinate, and

since the first coordinate of r(t1)i is ζbi/2c, we see that 〈r(t1)〉 ∩ 〈r(t2)〉 = 1. Since T+ is abelian,we then have

T+∼= 〈r(t1)〉 × 〈r(t2)〉 ∼= Z/(2n+ 2)Z× Z/n+1

2 Z,

where the isomorphism sends (i, j) ∈ Z/(2n+ 2)Z×Z/n+12 Z to ζbi/2cr(ti+2j) ∈ T+. From here on,

we will use this isomorphism freely.Since w−1

0 · r(ti) = ζir(ti), we see that

w−10 · (i, j) = w−1

0 · ζbi/2cr(ti+2j)

= ζbi/2c+i+2jr(ti+2j)

= ζb(3i+4j)/2cr(t(3i+4j)+2(−i−j)))

= (3i+ 4j,−i− j).

Also note that(3 4

−1 −1

)·

(2n+ 1 4n

−n −2n+ 1

)=

(2(n+ 1) + 1 4(n+ 1)

−(n+ 1) −2(n+ 1) + 1

),

so explicitly, w−`0 · (i, j) = ((2`+ 1)i+ 4`j, −`i+ (−2`+ 1)j).Now for a given character χ of A, write B for the maximal subgroup of C+ to which χ extends,

and write E for the image of this subgroup in Wϕs . As a subgroup of Wϕ

s , we know that

E ∼=⟨wk′

0

⟩∼= Z/n+1

k′ Z

for some k′|n+ 1. Define v := wk′

0 .

First, suppose that 2|k′. In this case, det(P (v)) = (−1)k′

= 1, so the map s : E → B sendingv` 7→ P (v`) gives a splitting of E into B. As a result, χ is virtually split.

On the other hand, suppose that 2 - k′. Since χ ∈ X(Z/(2n + 2)Z × Z/n+12 Z), we know that

χ(i, j) = xi1xj2, where x1 ∈ µ2(n+1) and x2 ∈ µ(n+1)/2. Since χ is fixed by E, we know in particular

that χ(1, 0) = v−1 · χ(1, 0). As a result,

x1 = x2k′+11 x−k

′

2 ,

so

(4.1.2) xk′

2 = x2k′1 .


Since det(ζ1/2P (w0)) = 1, the map s : E → B sending v` 7→ (ζ1/2P (w0))k′` gives a set-theoretic

section of E into B. The cocycle of this section is

ξs(vi, vj) =

1 i+ j < n+1

k′

−1 i+ j ≥ n+1k′ .

Note that −1 ∈ T+ corresponds to (n+ 1, 0) ∈ Z/(2n+ 2)Z×Z/n+12 Z. Since 2|n+ 1 but 2 - k′, we

know that 2|n+1k′ . Raising Equation 4.1.2 to the power of n+1

2k′ and remembering that x2 ∈ µ(n+1)/2,we see that

xn+11 = x

(n+1)/22 = 1.

As a result, χ ξs = 1, so in this case, χ is also virtually split. As a result, the sequence 1→ T+ →C+ →Wϕ

s → 1 is virtually split.

4.2. Type Bn. (1) Let

J =

0 · · · 0 1

0 1 0... . .

. ...

1 0 · · · 0

∈ Mat2n+1.

Then in this coordinate system,

Gad = SO2n+1 = X ∈ Mat2n+1 : det(X) = 1 and XTJX = Jand

Tad = t = diag(t1, .., tn, 1, t−1n , .., t−1

1 ) ∈ SO2n+1 : ti 6= 0.Then αn(t) = tn, αi(t) = ti/ti+1 if i 6= n gives the basis for ∆. The extended Dynkin diagram fortype Bn is

The alcove stabilizer Ω acts on D(g) via reflecting α0 and α1, so the elements fixed by a nontrivialreflection ω ∈Wϕ

s are given by

x =n∑i=0

sivi

such that s0 = s1,∑n

0 si = 1. In type Bn, v1 = ω1 and vi = ωi/2 if i 6= 1. Then a nontrivialω ∈Wϕ

s which fixes x isω = (1, 0, · · · , 0) o id ∈ (Z/2Z)n o Σn.

Furthermore, we seeWϕs = 〈ω〉 = Z/2Z,

and the regular, but not strongly regular elements are of the form

(4.2.1) s = diag(s1, · · · , sn, 1, s−1n , · · · s−1

1 ),

wheresi = e2πitj , and t1 = 1/2 > t2 > · · · tn > 0.

(2) Let w0 = τ oσ ∈ (Z/2Z)noΣn denote image of ϕ in W . Observe, Tϕ is finite if and only if w0

is a product of negative cycles. Furthermore, Wϕs 6= 1 if and only if ww0 = w0w, which implies

the first cycle of ϕ is length 1. Assume, after some conjugation, that ϕ = ϕi1 t · · · t ϕim where

i1 = 1, ϕik+1= (ik1 + 1, .., ik+1) is a negative cycle whose action on T is given by

ε∗ik+1 → ε∗ik+2, · · · , ε∗ik+1−1 → ε∗ik+1, ε∗ik+1

→ −ε∗ik+1.


With these restrictions on ϕ, we see

Tϕad∼= (Z/2Z)m,

where m is the number of (negative) cycles of ϕ.(3) Taking the general form of ϕ = τ o σ ∈ (Z/2Z)n o Σn and s = diag(s1, · · · , sn, 1, s−1

n , · · · s−11 )

where si = e2πitj , and t1 = 1/2 > t2 > · · · tn > 0 found in (1), (2), we see

w0.s = diag(− 1, s

τ(1)σ(1), s

τ(2)σ(2), · · · , s

τ(n)σ(n), 1, · · · ,−1

)= ϕ.s.ϕ−1,

and we want this to equal

sp =((−1)p, sp1, .., s

pn, 1, .., (−1)p

).

This places the restrictions

(4.2.2) tj =lj

pkj + 1, where kj = order of cycle in ϕ containing j

for lj integers which must make tj satisfy 1/2 = t1 > t2 · · · > tn > 0, meaning

τ(j)lσ(j) − pljpkj + 1

∈ Z.

The key takeaway is given a prime p and the rank(G)=n, there are only finitely many pairs (s, ϕ)which satisfy (1),(2),(3), and the restriction on s described in 4.2.2 allows for one to write acomputer program which outputs all possible pairs.

(4) We will show the Langlands-Shelstad section directly gives a splitting of Wϕs → Cent(s, ϕ; Gad).

The set of roots α ∈ Φ+ such that w−1α < 0 is given in the following table:

Table 4.2.1. Bn Roots

α > 0 ε1 ε1 − εj , j > 1 ε1 + εj , j > 1

w−1α < 0 −ε1 −ε1 − εj , j > 1 −ε1 + εj , j > 1

Using Table 2.3.1 and 4.2.1, we easily compute

n(w)2 = ξ(w,w) = 1/2[2ε∗1 +∑j>1

((ε∗1 + ε∗j ) + (ε∗1 − ε∗j ))] = nε∗1 ∈ P .

Recall Tad = V/P , so w2 = 1 ∈ Tad which shows the Langlands-Shelstad section is a splitting.Using that ϕ(1) = −1 and ϕ is product of negative cycles, we see

α ∈ Φ+ : w−1α < 0 and w−10 w−1α > 0 = ε1, ε1 ± εj , 1 < j,

and

α ∈ Φ+ : w−10 α < 0 and w−1w−1

0 α > 0 = ε1, ε1 ± εj , 1 < j.Since

[n(w), n(w0)] = ξs(w,w0)ξ(w0, w)−1 = nε∗1 + nε∗1 = 2ne∗1 ∈ Q,the lift of w lives in both Tϕad and T+. Finally, observe that w acts trivially on Tϕad, hence we haveshown

Cent(s, ϕ, Gad) = Tad ×Wϕs = (Z/2Z)m × Z/2Z,

where m is the number of cycles in ϕ.(5) We first prove

T+∼= (Z/4Z)× (Z/2Z)m−1.


We see a basis of Tϕad is given by 〈1/2ε∗1, 1/2ε∗2 + · · ·+ 1/2ε∗i2 , · · · , 1/2εim−1+1 + · · · 1/2εim〉. Thenthe pre-image in the simply-connected cover has basis given by 〈f, e2, .., em〉 where

f = 1/2ε∗1,

ek = 1/2ε∗ik+1 + · · ·+ 1/2ε∗ik+1if ik even ,

ek = 1/2ε∗1 + 1/2ε∗ik+1 + · · ·+ 1/2ε∗ik+1if ik odd .

We can see this indeed gives a basis and implies T+∼= Z/4Z× (Z/2Z)m−1. Now it is straigtforward

to verify w.f = f3 and w.ek = ek if ik even and w.ek = f3ek if ik odd.

By Table 4.2.1, ξ(w,w) ∈ Q if n is even, so we again have a splitting: Cent(s, ϕ; Gad)+∼=

T+ o Z/2Z and we know its representations. If n is odd, then n(w) has order 4. Let χ : Tϕ+ → C×

be a character. Then w.χ(f) = x1 = χ(w.f) = x31 implies x1 = ±1. Thus, χξs(w,w) = χ(−1) =

χ(f2) = 1 and thus from 3.2.4, we see σ : (Wϕs )χ = Wϕ

s must be a character. This means all

irreducible representations of Cent(s, ϕ; Gad)+ are either of the form

IndCent(s,ϕ;Gad)+

Tϕ+χ

when w.χ 6= χ or of the form

χ : Cent(s, ϕ; Gad)+ → C×,

defined by

χ(tn(w)) = χ(t)σ(w)

for a character σ of Wϕs .

4.3. Type Cn. Define the group homomorphism P : Σk → GLk(C) which sends an element σ ∈ Σk

to its corresponding permutation matrix P (σ), i.e. the matrix such that for all M ∈ Matk(C) wehave (P (σ)MP (σ)−1)ij = Mσ−1(i)σ−1(j). Also, given a function f : 1, . . . , k → C, define thematrix Λ(f) such that

Λ(f)ij =

f(i) i = j

0 i 6= j.

With these definitions, note that

P (σ)Λ(f)P (σ)−1 = Λ(f σ−1).

Now consider the alternating bilinear form

J =

0 0 · · · 0 1

0 0 · · · −1 0...

... . .. ...

...

0 1 · · · 0 0

−1 0 · · · 0 0

= Λ((−1)i)P (ρ) ∈ Mat2n(C),

where ρ ∈ Σ2n is the permutation sending i 7→ 2n + 1 − i. Having chosen this J , we can presentSp2n(C) as

Sp2n(C) = X ∈ GLn(C) | XTJX = J.In this presentation, we can choose the maximal torus

Tsc = diag(t1, . . . , tn, t−1n , . . . , t−1

1 ).

Then in the adjoint case we get

Gad = PSp2n(C) = Sp2n(C)/µ2


and

Tad = Tsc/µ2.

Now write b1, . . . , bn for the standard basis of (Z/2Z)n and define a homomorphism q1 :(Z/2Z)n → Σ2n which sends bi 7→ (i, 2n+ 1− i). Note that q1((Z/2Z)n) commutes with ρ. Thereis an embedding Σn → Σ2n given by having an element σ ∈ Σn act on only the first n coordinatesof 1, . . . , 2n. With this identification in mind, define the function q2 : Σn → Σ2n which sendsσ 7→ ρσρσ. Note that for any elements σ, τ ∈ Σn, we know that σ and ρτρ commute, so that ρcommutes with q2(σ) and q2 is a homomorphism. Finally, note that for any x ∈ (Z/2Z)n and any

σ ∈ Σn, we have that q2(σ)q1(x) = q1(σx), where Σn acts on (Z/2Z)n by permuting coordinates. Wetherefore obtain an injective group homomorphism q : W → Σ2n which sends xo σ 7→ q1(x)q2(σ).Note that q(W ) commutes with ρ.

For σ ∈ Σ2n, define the function sσ : 1, . . . , 2n → C by

sσ(i) =

(−1)i+σ

−1(i) i ≤ n1 i > n.

Note that if σ and ρ commute, we have

sσ ρ =

(−1)ρ(i)+σ−1ρ(i) ρ(i) ≤ n1 ρ(i) > n

=

(−1)i+σ

−1(i) i > n

1 i ≤ n

= (−1)i+σ−1(i)sσ.

Now define the map R : Σ2n → GL2n(C) which sends σ 7→ Λ(sσ)P (σ). Note that for σ ∈ Σρ2n

we have that

R(σ)TJR(σ) = P (σ−1)Λ(sσ)Λ((−1)i)P (ρ)Λ(sσ)P (σ)

= P (σ−1)Λ(sσ)Λ((−1)i)Λ(sσ ρ)P (ρσ)

= P (σ−1)Λ((−1)σ−1(i))P (ρσ)

= Λ((−1)i)P (ρ)

= J,

so R maps Σρ2n into Sp2n(C). As a result, n = R q is an injective map from W → Sp2n(C). Since

n(w) acts on our chosen Tsc in the same way as w, we see that n is a set-theoretic section of the

exact sequence 1→ Tsc → N(Tsc)→W → 1. Since n = R q, where q is a group homomorphism,we see that ξn = ξR q. Now we compute ξR:

ξR(σ, τ) = Λ(sσ)P (σ)Λ(sτ )P (τ)(Λ(sστ )P (στ))−1

= Λ(sσ)Λ(sτ σ−1)P (σ)P (τ)P (στ)−1Λ(sστ )

= Λ(sσ · (sτ σ−1) · sστ ).

Observe

sσ · sστ =

(−1)i+σ

−1(i)+i+τ−1σ−1(i) i ≤ n1 i > n

=

(−1)σ

−1(i)+τ−1σ−1(i) i ≤ n1 i > n


and

sτ σ−1 =

(−1)σ

−1(i)+τ−1σ−1(i) σ−1(i) ≤ n1 σ−1(i) > n,

so that

sσ · (sτ σ−1) · sστ =

(−1)σ

−1(i)+τ−1σ−1(i) (σ−1(i) ≤ n)⊕ (i ≤ n)

1 otherwise,

where here ⊕ denotes logical exclusive or. Now let m ∈ Σn denote the permutation sendingi 7→ n+ 1− i. Note that q1(m) is the permutation sending i 7→ n+ 1− i if i ≤ n and i 7→ 3n+ 1− iif i > n. If we suppose that σ and τ commute with q1(m), then we see that

(sσ · (sτ σ−1) · sστ ) q1(m) =

(−1)q1(m)σ−1(i)+q1(m)τ−1σ−1(i) (σ−1(i) ≤ n)⊕ (i ≤ n)

1 otherwise.

Note that q1(m)σ−1(i) + q1(m)τ−1σ−1(i) always equals σ−1(i) + τ−1σ−1(i) mod 2, so

(sσ · (sτ σ−1) · sστ ) q1(m) = sσ · (sτ σ−1) · sστ .We conclude that if we compose n with the projection p : Sp2n(C) → PSp2n(C), we obtain a

section of the exact sequence 1→ Tad → N(Tad)→W → 1 whose cocycle is p s.

(1) The highest root of a root system of type Cn is 2α1 + . . .+ 2αn−1 +αn so the vertices of thefundamental alcove are vi = ωi/2 for i < n and vn = ωn. The extended Dynkin diagram for Cn is

Hence Ω ∼= Z/2Z, where the nontrivial element acts by reflecting the diagram, i.e sending vi 7→ vn−i.As a result, when written in barycentric coordinates, the nontrivial elements of the fundamentalalcove which are fixed by this element are of the form

x =n∑i=0

sivi

such that si = sn−i,∑n

i=0 si = 1, and si > 0 for all i. Since ωi =∑i

j=1 ε∗j for i < n and

ωn =∑n

j=1 ε∗j/2, we see that

exp(x) = (eπit1 , . . . , eπitn , e−πitn , . . . , e−πit1),

where ti =∑n

j=i sj , so in particular ti + tn+1−i = 1 and ti > ti+1.

(2) Since any element ϕ ∈ W can be conjugated into an element which is a product of signedcycles, and since in Sp2n(C) it is obvious that a product of signed cycles has a finite set of fixed

points in Tsc if and only if it is a product of negative cycles, by 2.2.2 and 2.2.3 we see that Tϕ isfinite if and only if ϕ is conjugate to a product of negative cycles.

Write w for the projection of the nontrivial element of Ω to W . Then

w = (1, 1, . . . , 1) om.

Since Z/2Zn is abelian and (1, 1, . . . , 1) is fixed by every element of Σn under its action on Z/2Zn,we see that

w0ww−10 = w0(1, 1, . . . , 1)w−1

0 o w0mw−10 = (1, 1, . . . , 1) o w0mw

−10 ,

so w0 commutes with w if and only if w0 commutes with m. If it is the case that both w0

commutes with m and Tw0 is finite, then we can conjugate w0 by elements of W which commutewith m so that it is of the form emψmψ, where e sends ε∗dn/2e to −ε∗dn/2e when n is odd and ψ


is a product of negative cycles acting only on the coordinates 1, . . . ,⌊n2

⌋. Explicitly, there exist

1 ≤ i1 < i2 < . . . < im ≤⌊n2

⌋such that

ψ(ε∗j ) = ε∗j+1 for j /∈ i1, i2, . . . , im and ψ(ε∗ij ) = −ε∗ij−1+1.

On the other hand, if we can conjugate w0 into such a form by elements of W which commute with

m, then Tϕ is finite and ϕ commutes with m. As a result, we will now assume that w0 is of theform emψmψ.

Write l for the number of cycles of φ. We will now show that

T φad∼=

(Z/2)l all cycles have even length or all have odd length

(Z/2)l−2 × Z/4 otherwise.

Since T φsc ∼= (Z/2)l, we know that |T φad| = 2l. We will write out a basis for T φad to describe itexplicitly. For notational convenience, we will present basis elements via elements of V whichexponentiate to desired basis element. If w0 has at least one cycle of even length and one of oddlength, then we can conjugate w0 so that the first cycle has odd length and the second cycle has

even length. In this case, if n is even, a basis for T φad is given by e3, . . . , em, em′ , . . . e1′ , f, where

ek =1

2ε∗ik + . . .+

1

2ε∗ik+1−1,

ek′ =1

2ε∗i′k+1+1 + . . .+

1

2ε∗i′k,

f =∑

ik+1−ik even

ik+1−ik−1∑j=0j even

1

2ε∗ik+j +

1

2ε∗i′k−j

+∑

ik+1−ik odd

ik+1−ik−1∑j=0

(−1)j

4ε∗ik+j +

(−1)j

4ε∗i′k−j

.

Here, i′k denotes m(ik) = n + 1 − ik. Note that φekφ−1 = ek and φfφ−1 = −f = f , so these

elements all lie in T φad. We know ek has order 2, and since there is at least one even cycle andat least one odd cycle, f has order 4. Because all of the basis elements we have written down

are linearly independent, we see by a counting argument that they span T φad. As a result, in this

case T φad∼= (Z/2)l−2 × Z/4. The case when n is odd is only slightly different: just take the basis

e3, . . . , em, em+1, em′ , . . . e1′ , f, where em+1 = 12ε∗dn/2e.

On the other hand, if either all of the cycles of w0 have odd length, or if they all have evenlength, then f has order 2. In this case, if n is even, take the basis e2, e3, . . . , em, em′ , . . . e1′ , fand if n is odd, take the basis e2, e3, . . . , em, em+1, em′ , . . . e1′ , f. Since all basis elements have

order 2, we see T φad∼= (Z/2)l.

(3) Assume s is as given in (insert equation) and w0 is in the normal form discussed in step (2).Let w−1

0 .(s1, . . . , sn) = (s2, s3, . . . , si1 , s−11 , . . . ), so we will actually compute pairs (s, φ) : φ−1sφ =

sp because it makes the notation easier. We follow the same procedure as in step (3) for type

An, Bn to find s = (s1, . . . , sn) must satisfy spi1+1

1 = εi1+1, where ε2 = 1 and we get a string of


inequalities:

1/2 >k1(i1 + 1)

2pi1+1>

pk1(i1 + 1)

2(pi1 + 1)

> · · · >

pi1−1k1(i1 + 1)

2(pi1 + 1)

>

...

>kj(ij − ij−1 + 1)

2(pij−ij−1 + 1)>

pkj(ij − ij−1 + 1)

2(pij−ij−1 + 1)

> · · · >

pij−ij−1−1kj(ij − ij−1 + 1)

2(pij−ij−1 + 1)

>

...

> 1/4,

where kj ∈ Z, j ∈ 1, . . . ,m,m = number of cycles of w0, are the free parameters chosen to makethis inequality hold, and x denotes the fractional part of x. The key takeaway again is for agiven prime p, there exists only finitely many pairs (s, φ) which can be determined systematicallyon a case-by-case basis using a computer program.(4) In order to apply the machinery of section (3.4) to n, we need to pick lifts of c(id, w0), c(w,w0),ξ(1, 1), ξ(1, w), ξ(w, 1), and ξ(w,w). Lift c(id, w0), ξ(1, 1), ξ(1, w), and ξ(w, 1) to 0. Note that

ξs(w,w) =

(−1)w(i)+i (w(i) ≤ n)⊕ (i ≤ n)

1 otherwise

= Λ((−1)w(i)+i)

= (−1)n,

so we can lift ξ(w,w) to nωn. Finally, note that c(w,w0) = Λ((−1)f(i)), where f : 1, . . . , 2n →0, 1. We can then lift c(w,w0) to

∑ni=1 f(i)x∗i /2. Since both w0 and w commute with m, we know

that m · c(w,w0) = c(w,w0), so f m = f . From this, we can see that w · c(w,w0) = −c(w,w0).As a result, for any w1, w2 ∈Wϕ

s , we see that

c(w1, w0) + w1 · c(w2, w0)− c(w1w2, w0) = 0.

We can then conclude that

c(w1, w0) + w1 · c(w2, w0)− c(w1w2, w0) + (ϕ− 1)ξ(w1, w2) = (ϕ− 1)ξ(w1, w2)

is equal to (ϕ − 1)p for some p ∈ P (either 0 or nωn). As a result, the sequence 1 → Tϕ →Cent(s, ϕ; Gad)→Wϕ

s → 1 splits.

(5) First we will show that

T+∼=

(Z/2)l+1 all cycles of w0 have even length

(Z/2)l−1 × Z/4 otherwise,

where l is the number of cycles of w0. First note that T+ is the pullback of T φad along a map with

kernel of size 2, so |T+| = 2l+1. Using the same notation as in part 2, if all of the cycles of w0 have

even length then we can pick a basis for T+ given by e1, . . . , em, em′ , . . . , e1′ , f (Whereas before,ek was the image of 1

2ε∗ik

+ . . .+ 12ε∗ik+1−1 under the exponential map from V to PSp2n, it is now the

image of the same element of V under the exponential map from V to Sp2n). All of these elements

lie in T+ and they all have order 2. Since they are independent, we see by a counting argument

that they generate T+. As a result, T+∼= (Z/2)l+1.

On the other hand, if w0 has at least one cycle with odd length, then we can conjugate w0 sothat this is the first cycle of w0, and take the basis e2, . . . , em, em′ , . . . , e1′ , f when n is even and


e2, . . . , em, em+1, em′ , . . . , e1′ , f when n is odd. In this case, f has order 4, while all other basis

elements have order 2. As before, all of these basis elements lie in T+, and they are all independent,

so they generate T+. As a result, we see that T+∼= (Z/2)l−1 × Z/4.

Now note that writing ϕ = y o σ for y ∈ Z/2Zn and σ ∈ Σn, we see that ϕ · ωn = y · ωn, andif we (again employing the standard basis for Z/2Zn) write y =

∑i∈I bi for some I ⊂ 1, . . . , n,

then y · ωn = ωn −∑

i∈I x∗i . As a result, (ϕ− 1)ωn = −

∑i∈I x

∗i . Now consider p ∈ V given by

p =∑i∈I

x∗i2

+∑i/∈I

i≤bn/2c

x∗i − x∗m(i)

2.

Since ϕ commutes with m, we know that I is preserved by m, so

w · p = −∑i∈I

x∗i2

+∑i/∈I

i≤bn/2c

x∗i − x∗m(i)

2,

so that (w − 1)p = (ϕ − 1)ωn. Since (p + ωn) ∈ Zx∗1, . . . , x∗n, we see that p ∈ P . Because

(ϕ− 1)ξ(w1, w2) is always some integer multiple of ωn, we can then conclude that

c(w1, w0) + w1 · c(w2, w0)− c(w1w2, w0) + (ϕ− 1)ξ(w1, w2) = (w − 1)p′

for any w1, w2 and some corresponding p′ ∈ P . As a result, the sequence 1→ T+ → Cent(s, ϕ; Gad)+ →Wϕs → 1 is virtually split.

4.4. Type Dn. Since Ω = Z/2Z × Z/2Z if n is even and Z/4Z if n is odd, we split into cases ofparity of n.

Case D2n

(1) Let

J =

0 · · · 0 1

0 1 0... . .

. ...

1 0 · · · 0

∈ Mat4n.

Then in this coordinate system,

Gad = PSO4n = (X ∈ Mat4n : det(X) = 1 and XTJX = J)/µ2

and

Tad = t = diag(t1, .., t2n, t−12n , .., t

−11 ) ∈ PSO4n : ti 6= 0.

Then α2n(t) = t2n−1 + t2n, αi(t) = ti/ti+1 if i 6= n gives the basis for ∆. The extended Dynkindiagram for type D2n is

The non-trivial symmetries of this Dynkin diagram are

Ω = 〈w1〉 × 〈w2〉 = Z/2Z× Z/2Z,

where

w1 : αj ↔ α2n−j for all 0 ≤ j ≤ 2n

and

w2 : α0 ↔ α2n−1 α1 ↔ α2n αj ↔ α2n−j for 2 ≤ j ≤ 2n− 2.


In type D2n, v1 = ω1, vi = ωi/2 for i = 2, .., 2n− 2, v2n−1 = ω2n−1, v2n = ω2n. So the regular butnot strongly regular elements are ones fixed by a subgroup of 〈w1〉 × 〈w2〉. We treat the nontrivialcase when it is fixed by both w1, w2 so that Wϕ

s is not cyclic. Then

x =

2n∑i=0

sivi,

where s0 = s1 = s2n−1 = s2n and si = s2n−i for 2 ≤ i ≤ 2n− 2. Thus

exp(x) = diag(e2πit1 , · · · e2πit2n , e−2πit2n , · · · , e−2πit1),

where t1 = 1/2 > t2 > · · · > t2n−1 > t2n = 0 and ti + t2n−i+1 = 1/2 for all i. We can now describe

the w1, w2 action on T : w1 = e1e2n and w2 = (−1).m, where ei : ε∗i → −ε∗i and fixes ε∗j for j 6= iand m : ε∗j → ε∗2n+1−j for all j.

(2) In order for w0 to commute with w1, w2 and to make Tϕad finite, it is shown, in the sameway as for type Bn, Cn, that w0 must be conjugate to a “normal form” w′0 · mw′0m−1, wherew′0 = ϕi1 t · · ·ϕim , 1 = i1 < 2 = i2 < i3 < · · · < im < im+1 := n + 1, is a signed permutation in nvariables with action on ε∗1, · · · ε∗n given by

(w′0)−1 : ε∗j → ε∗j−1for j /∈ i1, i2, · · · im and (w′0)−1 : ε∗ia → −ε∗ia+1−1.

We further say w′0 acts as identity on e∗n+1, · · · , ε∗2n to make this an action on Tϕad. Notice this ϕ

has the same normal form as that for type Cn, so Tϕad is the same:

(4.4.1) Tϕad =

(Z/2Z)m if all cycle lengths ik+1 − ik are odd

(Z/2Z)m−2 × Z/4Z else.

(3) The restrictions that φsφ−1 = sp makes on (s, φ, p) for type Dn are very similar to that of typeBn and Cn. We omit it for that reason and invite the reader to attempt it. Let us know if you finda (nicer) characterization!(4) Let us record the tables arising while computing the Langlands-Shelstad section:

Table 4.4.1. Ws action on D2n Roots

α ε1 ± ε2n ε1 ± εj , j < 2n εi ± ε2n, i > 1 εi ± εj , 1 < i < j < 2n

w−11 α −(ε1 ± ε2n) −(ε1 ∓ εj) εi ∓ ε2n εi ± εj

α εi − εj , i < j εi + εj , i < j

w−12 α ε2n+1−j − ε2n+1−i −(ε2n+1−j + ε2n+1−i)

For the action of w−10 , we only need to record which positive roots get sent to negative. Let

B = ia : a = 1, .., k ∪ i′a : a = 1, .., k,where i′a = 2n+ 1− ia.

We compute n(w1)2 = ξs(w1, w1) = 2nε∗1 ∈ Q and n(w2)2 = 1/2((2n−1)ε∗1 + ..+(2n−1)ε∗2n) ∈ Pshows both lifts have order two in Tad.

Now, consider the lift the section into the centralizer as described in (3.4.1). Then s(w1) =tw1n(w1), s(w2) = tw2n(w2), and since we are using the Langlands-Shelstad lift, the commutatorc(w2, w0) is represented by the half sum of the coroots for all roots occurring in Λw2,w0 , where

Λu,v := α > 0, (uv)−1α > 0⋂(u−1α < 0, v−1α > 0 ∪ u−1α > 0, v−1α < 0

).


Table 4.4.2. w−10 action on D2n roots

α w−10 (εi − εj), i < j w−1

0 (εi + εj , i < j)

i, j /∈ B + +

i = ia, j < ia+1 − +

i > i′a+1, j = i′a + −i = ia, j ≥ ia+1 − −

i = i′a − −ia < i < ia+1, j ≥ ia+1 + +

i′a+1 < i < i′a, j > i′a + +

Using tables 4.4.1, we find Λw1,w0 = ø, hence n(w1) is Ad-φ fixed and tw1 = 1. Next we computeΛw2,w0 :

Λw2,w0 =εi ± εj : i = ia, j < ia+1∪ εi − εj : i = ia, j ≥ ia+1, j /∈ B∪ εi − εj : i = i′a, j /∈ B∪ εi + εj : j = ia, i /∈ B∪ εi + εj : j = i′a, i ≤ i′a+1, i /∈ B.

Since we chose the section to be the Langlands-Shelstad section, the lift of

c(w2, w0)(−1) =∏

α∈Λw2,w0

α(−1) ∈ T

to V is represented by1

2c(w2, w0) =

1

2

∑α∈Λw2,w0

α.

To compute c(w2, w0), we check the contributions for ε∗ib and ε∗i for ib < i < ib+1:

c(w2, w0)|ib =∑

ib<j<ib+1

[(ε∗ib + ε∗j ) + (ε∗ib − ε∗j )]+∑

j≥ib+1

j /∈B

(ε∗ib − ε∗j ) +

∑i/∈Bi<ib

(ε∗i + ε∗ib),

where c|i denotes restrict summation to terms including ε∗i . So the coefficient for ε∗ib is 2(ib+1 −ib − 1) + (2n− ib+1 + 1− (|B| − b)) + (ib − 1− (b− 1)) = ib+1 − ib + 2n− |B| − 1

By doing the same thing for ε∗i′b, we see its coefficient is ib − ib+1 + 2n− |B|+ 1.

Next, we consider ib < i < ib+1:

c(w2, w0)|i =∑

a:i<ia+1

(ε∗ib ± εi) +∑

a:i≥ia+1

i/∈B

(ε∗ia − ε∗i )+

∑a:i<iai/∈B

(ε∗i + ε∗ia) +∑

a:i≤i′a+1

i/∈B

(ε∗i + ε∗i′a).

So the coefficient of ε∗i is (1− 1)− (b− 1) + (k − b) + k = |B| − 2b+ 1.


In much a similar way, one finds coefficient for ε∗i , i′b+1 < i < i′b is 2b− |B| − 1. In conclusion,

C(w2, w2)i := 1/2(w2.c(w2, w0) + c(w2, w0))i =

ib+1 − ib − 1 if i = ib

−ib+1 + ib + 1 if i = i′b2k − 2b+ 1 if ib < i < ib+1

−2k + 2b− 1 if i′b+1 < i < i′b,

where Ci denotes the ith coordinate of C in the ε∗i coordinates.

Finally, it is immediate to check C(w2, w2) = (φ− 1)p, where p ∈ P is defined as p = (pi), withpib+1−1 = (ib+1 − ib − 1)(k − 1), and pib−1−l = pib−1 + l(2k − 2b+ 1) for 1 < l < ib+1 − ib. Thus we

have shown s(w2)2 = (tw2n(w2))2 = 1.

To summarize: s(w1) = n(w1), s(w2) = tw2n(w2) and the computation for C(w2, w2) showss(w2)2 = (twn(w))2 = 1. Observe [s(w1), s(w2)] = (t−1

w2·w1tw1)[n(w1), n(w2)] and Λw1,w2 = ε1±εj :

1 < j < 2n implies 1/2c(w1, w2) = 1/2(2n−2)(ε1+ε2n) ∈ Q, thus [n(w1), n(w2)] = 1. We computedirectly (t−1

w2·w1 tw1) = |B|/2ε1+|B|/2ε2n ∈ Q, thus (t−1

w2·w1tw1) = 1 and [s(w1), s(w2)] = 1. Finally,

s(w1)2 = n(w1)2 = 2nε1 ∈ Q. This all shows s is a splitting map, so Cent(s, ϕ; Gad) = T φad oW φs .

(5) First, we will show that

T+∼= (Z/2)l−2 × (Z/4)2,

where l is the number of cycles of w0. First note that since |T φad| = 2l and since |Z(Gsc)| = 4, we

know that |T+| = 2l+2. Now we give an explicit basis for T+: f1, e2, . . . ek, ek′ , . . . , e2′ , f2, where

f1 =1

2ε∗1,

eb =

12ε∗ib

+ . . .+ 12ε∗ib+1−1 ib+1 − ib even

12ε∗ib

+ . . .+ 12ε∗ib+1−1 + 1

2ε∗2n ib+1 − ib odd,

f2 =∑

ib+1−ib even

ib+1−ib−1∑j=0j even

1

2ε∗ib+j +

1

2ε∗i′b−j

+∑

ib+1−ib odd

ib+1−ib−1∑j=0

(−1)j

4ε∗ib+j +

(−1)j

4ε∗i′b−j

,and eb′ = m(eb). These elements are all in T+, and they are all linearly independent. Furthermore,

f1 and f2 have order 4, while eb has order 2 for all b. As a result, these elements generate T+, so

we know T+∼= (Z/2)l−2 × (Z/4)2.

We find

(w0 − 1)ξ(w2, w2)i =

−(2n− 1) if i ∈ B0 else,

so take p′i =

2n−1

2 if i ∈ B2n−1

2 if i /∈ B, i ≤ n−2n−1

2 if i /∈ B, i > n.

The p coming from C = (w0 − 1)p is actually in Q, so p = q. Then A(w2, w2) = C(w2, w2) +

(w0 − 1)ξ(w2, w2) = (w0 − 1)q + (w2 − 1)p′ for some q ∈ Q, p′ ∈ P , which shows Cent(s, ϕ; Gad)+

is virtually split.Case D2n+1


(1) Notes: Ω = Z/4Z = 〈w〉, w acts on extended Dynkin Diagram of type D2n+1, by

α0 → α2n+1 α2n+1 → α1

α1 → α2n α2n → α0

αj ↔ α2n+1−j for 2 ≤ j ≤ 2n− 1.

(Note: there is a typo in [Bou02, Plate IV]) We assume a regular element is fixed by all of Ω = 〈w〉.Then one finds it is of the form:

x =2n∑i=0

sivi,

where s0 = s1 = s2n = s2n+1 and si = s2n+1−i for 2 ≤ i ≤ 2n− 1. Thus

exp(x) = diag(e2πit1 , · · · e2πit2n+1 , e−2πit2n+1 , · · · , e−2πit1),

where t1 = 1/2 > t2 > · · · > t2n > t2n+1 = 0 and ti + t2n−i+2 = 1/2 for all i. Then w acts on T by

ε∗i → −ε∗2n+2−i for i < 2n+ 1, ε∗2n+1 → ε∗1.

(2) For the same reasons as in Cn, we see that

T φad∼= (Z/2)l−2 × Z/4.

The normal form for w0 is similar to that for D2n, with the exception of the action on the ε1, ε2n+1

coordinates. Let i1 = 1 < i2 = 2 < · · · < im < im+1 := n+ 1 < im+2 := n+ 2. Define the action

(w′0)−1 : εi → εi−1, if i 6= i2, . . . , im and εib → −εin+1−1,

en+1 : εn+1 → −εn+1,

(e′1)−1 : ε1 → ε2n+1, ε2n+1 → −ε1.

Then w0 = e′1en+1w′0mw

′−10 is the normal form that we will consider below.

(3) The restrictions that φ.sφ−1 = sp makes on (s, φ, p) for type Dn are very similar to that oftype Bn and Cn. We omit it for that reason and invite the reader to attempt it. Let us know ifyou find a (nicer) characterization!(4)

Table 4.4.3. Ws action of D2n+1 roots

α ε1 − εj ε1 + εj εi − εj , 1 < i εi + εj , 1 < i

w−1α ε2n+1 + ε2n+2−j ε2n+1 − ε2n+2−j ε2n+2−j − ε2n+2−i −(ε2n+2−j + ε2n+2−i)

w2α − − + +

We first show s(w)2 = s(w2), or equivalently

1/2(w0 − 1)−1[c(w,w0) + w.c(w,w0)− c(w2, w0) + ξ(w,w)] ∈ P ,

and then we finish by showing s(w2) has order 2 in Tad.We find

n(w)2 = ξ(w,w) = (2n− 1)/2ε∗2 + (2n− 1)/2ε∗3..+ (2n− 1)/2ε∗2n+1.


Thus,

1/2(φ− 1)ξ(w,w)i =

2n−1

2 if i = 1

−(2n− 1) if i ∈ B′′ or i = n+ 1

−2n−12 if i = 2n+ 1

0 else.

Let B′′ = i2, .., ik, i′2, .., i′k, B′ = B′′ ∪ n + 1, B = B′ ∪ 2n + 1, ik+1 := n + 1, ik+2 :=n + 2, ε0 := ε2n+1. Then the D2n+1 table is the union of the subset of the D2n table for B′′ inV ′′ = spanε2, · · · , εn, εn+2, · · · ε2n (so the w−1

0 for D2n+1 coincides with the w−10 from D2n when

restricted to V ′′) and the following exceptions:

Table 4.4.4. w−10 action on D2n+1 (exception) roots

α w−10 (εi − εj) w−1

0 (εi + εj) w−1w−10 (εi − εj) w−1w−1

0 (εi + εj)

i = 1, j /∈ B − + − −i = 1, j ∈ B, j 6= 2n+ 1 + − − −

i = 1, j = 2n+ 1 + − − −i /∈ B, i 6= 1, j = 2n+ 1 + − − −

i ∈ B, j = 2n+ 1 + − + +

i = n+ 1, j /∈ B − − + +

i = n+ 1, j ∈ B, j 6= 2n+ 1 − − − +

i /∈ B, i 6= 1, j = n+ 1 + + − −i ∈ B, j = n+ 1 − − + +

We conclude

Λw2,w0=ε1 + εj , j /∈ B ∪ ε1 − εj , j ∈ B, j 6= 2n+ 1∪ εi + ε2n+1, i /∈ B, i 6= 1 ∪ εi + ε2n+1, i ∈ B.

Thus

1/2c(w2, w0)i = [1/2∑

α∈Λw2,w0

α]i =

(2n− 1)/2 if i = 1, 2n+ 1

0 if i ∈ B′′ or i = n+ 1

1 if i /∈ B, i 6= 1,

where as usual, ci denotes ith coordinate of c in ε∗i basis. Thus,

(4.4.2) − 1/2c(w2, w0) + 1/2(φ− 1)ξ(w,w)i =

(2n− 1) if i = 1

0 if i = 2n+ 1

−(2n− 1) if i ∈ B′′ or i = n+ 1

1 else.

We further find

Λw,w0 =εi ± εj : i = ia ∈ B′, j < ia+1∪ εi − εj : i = ia, j ≥ ia+1, j /∈ B′∪ εi − εj : i = i′a, j /∈ B′∪ εi + εj : j = ia, i /∈ B′∪ εi + εj : j = i′a, i ≤ i′a+1, i /∈ B′.


This implies,

(4.4.3) C(w,w) := 1/2(w.c(w,w0) + c(w,w0))i =

0 if i = 1, n+ 1, 2n+ 1

ib+1 − ib − 1 if i = ib ∈ B′′

−ib+1 + ib + 1 if i = i′b ∈ B′′

2k − 2b+ 1 if ib < i < ib+1

−2k + 2b− 1 if i′b+1 < i < i′b.

We see A is the sum of equations (4.4.2), (4.4.3) and it is not equal to (w0 − 1).p for any

p ∈ P , which shows s is not a section. However, we still do have the decomposition A(w,w) =(w0 − 1)q + (w2 − 1)p by the following. Equation 4.4.2 is of the form (w − 1)p for p = (pi) and

(4.4.4) pi =

−(2n− 1)/2 if i = 1

(2n− 1)/2 if i = n+ 1, 2n+ 1, or i ∈ B′′

−1/2 else.

Equation 4.4.3 is of the form (w0 − 1)q for q ∈ Q for the same reason C occurring in D2n was of

this form. This shows that Cent(s, ϕ; Gad) is virtually split.

(5) We will show that

T+∼= (Z/2)l−1 × Z/8,

where l is the number of cycles of w0. First note that since |Z(Gsc| = 4 and |T φad| = 2l, we must

have |T+| = 2l+2. Now we give an explicit basis for T+: f1, e2, . . . , ek, ek′ , . . . , e2′ , f2, where

f1 =1

2ε∗1 +

1

2ε∗2n+1,

eb =

12ε∗ib

+ . . .+ 12ε∗ib+1−1 ib+1 − ib even

12ε∗ib

+ . . .+ 12ε∗ib+1−1 + 1

2ε∗n+1 ib+1 − ib odd,

f2 =∑

ib+1−ib even

ib+1−ib−1∑j=0j even

1

2ε∗ib+j +

1

2ε∗i′b−j

+∑

ib+1−ib odd

ib+1−ib−1∑j=0

(−1)j

4ε∗ib+j +

(−1)j

4ε∗i′b−j

.and eb′ = m(eb). These are all elements of T+, and they are all linearly independent. Furthermore,

f2 has order 8, while f1 and eb have order 2. As a result, these elements generate T+, and so

T+∼= (Z/2)l+1 × Z/8.

Cent(s, ϕ; Gad)+ is also virtually split for the same reason it is virtually split in the adjoint case.


References

[Bor79] A. Borel - Automorphic L-functions, Automorphic forms, representations and L-functions (Proc. Sympos.Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII,Amer. Math. Soc., Providence, R.I., 1979, pp. 2761. MR 546608 (81m:10056)

[Bro82] K. Brown - Cohomology of Groups. Spring-Verlag, 1982, ISBN 978-1-4684-9329-0[Bou02] N. Bourbaki - Lie groups and Lie algebras. Springer, 2002, ISBN 3-540- 42650-7[DR09] S. DeBacker, M. Reeder - Depth-zero supercuspidal L-packets and their stability. Ann. of Math. (2) 169

(2009)[Gel84] S. Gelbart - An elementary introduction to the Langlands program - Bulletin of the AMS, vol 10, no 2,

1984[Hum70] J. Humphreys - Introduction to Lie algebras and their representations. Springer-Verlag, 1970, ISBN 0-

387-90053-5[Kal15] T. Kaletha -The local Langlands conjectures for non-quasi-split groups. Families of Automorphic Forms

and the Trace Formula, Simons Symposia, Springer 2016, 217-257.[Kal16] T. Kaletha - Regular supercuspidal representations, 2016[Kal18] T. Kaletha - Supercuspidal L-packets, Appendix (draft)[LS87] R.P. Langlands, D. Shelstad - On the definition of transfer factors. Math. Ann., vol. 278 (1987), 219-271[R09] M. Reeder - Torsion automorphisms of simple Lie algebras. Enseign. Math.(2) 56.1-2 (2010): 3-47[Ser71] J.P. Serre - Linear representations of finite groups. Springer-Verlag, 1971, ISBN 0-387-90190-6.[Yu01] Yu, Jiu-Kang - Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), no.

3, 579622 (electronic). MR 1824988 (2002f:22033)

Department of Mathematics, University of California, Berkeley,E-mail address: [email protected]

Department of Mathematics, University of Texas, Austin,E-mail address: [email protected]

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