Journal of Lie Theory Volume 10 (2000) 269–284 ◯^{C}2000 Heldermann Verlag

On finite dimensional representations of non-connected reductive groups

David Joyner

Communicated by A. Valette

Abstract. We extend the classification of irreducible finite dimensional representations of almost simple algebraic groups over an algebraically closed field of characteristic zero to certain non-connected groups G where the component group is cyclic. We also extend some of Steinberg’s results on the adjoint quotient G → T/W to these non-connected groups. These results are used to describe the geometry of θ-conjugacy classes of G^{o}, where θ is an automorphism of the connected group G^{o}. As an application, we show that there is a “functorial” correspondence between virtual (finite dimensional) characters of θ-invariant representations of G and virtual characters of an endoscopic group H of G.

Contents

1. Introduction 269

2. Background on conjugacy classes 270

3. Some representation theory 272

4. θ-conjugacy classes 277

5. Some corollaries 278

6. Character relations and endoscopic groups 279

7. Irreducible representations of \overline{G}, G simple 281

1. Introduction

Let k denote an algebraically closed field of characteristic zero. Unless stated otherwise G will always denote a connected semisimple (linear algebraic) group

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag over k. In this paper, we extend some results of Steinberg to non-connected groups. Let θ denote an (algebraic) automorphism of G of finite order which fixes a “splitting” of G (this will be defined later). Let G⋊ < θ > denote the semidirect product where the multiplication is given by (g, σ)(g′, σ′) = (gσ(g′), σσ′)). G may be identified with a normal subgroup of \overline{G}.

In one of our main results, we describe a twisted analog of the “adjoint quotient G → T/W ”, where T is a Cartan of G and W is its Weyl group, as well as prove some basic facts regarding the geometry of “twisted” conjugacy classes. As a corollary to this, we prove that a θ-conjugacy class of G is Zariski closed if it is θ-semisimple. As a related matter, we investigate the characters of the irreducible finite dimensional representations of non-connected semisimple groups of the form G⋊ < θ >. In another of our main results, we prove that there is a natural 1-1 correspondence between the irreducible finite dimensional representations of G⋊ < θ > and those representations on a related “endoscopic” group (one must be careful in interpreting this statement - see Remark 3.3 below).

In general, if X denotes an algebraic variety defined over k then we often identify X with X(k).

2. Background on conjugacy classes

Let G be a closed connected (algebraic) subgroup of GL(V), where V is a finite dimensional k-vector space, and let θ be an algebraic automorphism of G of finite order. Denote the semidirect product described above by \overline{G} = G⋊ < θ >. Denote the connected components of \overline{G} by G = G.1, G.θ, G.θ^{2}, .... Without loss of generality (taking V to be the Lie algebra of G), we can (and do) assume that \overline{G} is also a subgroup of GL(V).

Lemma 2.1. Let \overline{G} be as above and let X denote a variety defined over k on which \overline{G} acts. Then

(a) every orbit is open in its closure,

(b) for each x ∈ X, the (Zariski) closure \overline{G · x} of an orbit G · x is a union of G · x and other orbits of smaller dimension,

(c) orbits of minimal dimension are closed.

For the proof in the connected case, see [10], §1.13.

Proof. Since θ acts on X, it is an isomorphism of X to itself. In particular, the set of orbits of G.θ^{i} on X is the same as the set of orbits of G.1 ≅ G on X. Therefore the Lemma above is a consequence of the connected case.

Proposition 2.2. (Jordan decomposition) Let \overline{G} be as above. Each g ∈ \overline{G} has a decomposition g = su = us, where s ∈ \overline{G} is semisimple (as an element of \overline{G} or, equivalently, as an endomorphism on V ) and u ∈ \overline{G} is unipotent (as an element of \overline{G} or, equivalently, u − 1 is a nilpotent endomorphism on V ). Furthermore, g determines s and u uniquely.

For the proof, see [10], §§2.1-2.4. As a consequence, we have the following result.

Corollary 2.3. Let G be as above. Let θ denote an automorphism of G of finite order. Then any g.θ ∈ \overline{G} has a decomposition

g.θ = (s.θ)u = u(s.θ),

for some unique s.θ semisimple in \overline{G} and u unipotent. Equivalently,

g = sθ(u) = us.

Definition 2.4. In the above corollary, we write g_{θ−ss} = s and g_{θ−un} = u. We call g θ-semisimple if g.θ ∈ \overline{G} is semisimple (i.e., if g = g_{θ−ss}). We call g θ-unipotent if g_{θ−ss} = 1, so g = θ(u) = u for some unipotent u.

We call x, y ∈G θ-conjugate if and only if y = g^{−1}xθ(g) for some g ∈ G.

Remark 2.5. The map

g ⟼ g_{θ−ss}

is not a morphism [8], §3.3. However, later we will construct a morphism (the twisted analog of the adjoint quotient) which is closely related to this map.

A Borel pair of a connected reductive group G is a pair (B, T), where T is a maximal torus contained in a Borel subgroup B of G. An automorphism of G which preserves a Borel pair (B, T) will be called quasi-semisimple. Let W denote the Weyl group of T and W^{θ} denote the θ-invariant elements of W.

Lemma 2.6. Let G denote a connected reductive group over k. Assume that θ is a quasi-semisimple automorphism of G which is of finite order and fixes a Borel pair (B, T) of G. Then

N_{G}(T.θ)/T = W^{θ},

where W s denote the Weyl group of T.

Proof. Write the Bruhat decomposition as G = BW B (where we have temporarily identified W with a complete set of coset representatives of W in N_{G}(T)). The lemma follows from the claim: if w ∈ W and g = bwb′ ∈ BwB satisfies gT θ(g)^{−1} = T then g ∈ T wT and w ∈ W^{θ}. We next prove this claim.

Let B = NT = T N, where N is the unipotent radical of B. Note that since θ preserves T and B, it preserves N. Suppose that g = tnwn′, for n, n′ ∈ N and t ∈ T. The equation gT θ(g)^{−1} = T implies θ(g)T θ(g)^{−1} = θ(g)g^{−1}T. The theory of maximal tori implies gθ(g)^{−1} ∈ T and θ(g) ∈ N_{G}(T). In turn, this implies g ∈ N_{G}(T) and that the image of θ(g)g^{−1} in W is the identity. This (by the Bruhat decomposition) implies the claim, from which the lemma follows.

For x ∈ G, let
C_{θ}(x) = {g^{−1}xθ(g) | g ∈ G},

the θ-conjugacy class of x. A splitting of G is a triple (B, T, {X}), where (B, T) is a Borel pair of G and {X} is a set of root vectors of T, one for each simple root of T in B. Let W and W^{θ} be as in the previous lemma. Let T_{θ} = T/(1 − θ)T denote the group of coinvariants of T.

The lemma below, based on the work of Steinberg [11], is Lemma 3.2.A in Lemma 2.7. Let G denote a connected reductive group over k and assume that the automorphism θ is of finite order and fixes a splitting ^{1} (B, T, {X}) of G. If s ∈ G is θ-semisimple then

(a) each O = C_{θ}(s) meets T,

(b) the image of O ∩ T in T_{θ} is a single W^{θ} -orbit.

Remark 2.8. This implies that there is a bijection ^{2}

G_{θ−ss} → T_{θ}/W^{θ}

(essentially the abstract norm map defined in in [4] §3.2) from the set of conjugacy classes of θ-semisimple elements of G to the quotient T_{θ}/W^{θ}. The construction of the above bijection should be compared with Corollary 5.4 below, which obtains this mapping from a character-theoretic perspective.

3. Some representation theory

Let G be a connected semisimple algebraic group over k. Let θ be an quasisemisimple automorphism of G. Let \overline{G} be as in the previous section.

If ρ is a finite dimensional representation of G, let

ρ^{θ}(g) = ρ(θ(g)), g ∈ G,

so ρ^{θ} is irreducible if and only if ρ is. We call a representation ρ θ-invariant if ρ^{θ} ≅ ρ. The purpose of this section is to classify such representations.

We denote the simple roots of (B, T) by Δ, the root system of (B, T) by R = R(B, T), the root lattice of T ⊂ G by Q = ℤ[Δ] = ℤ[R], the character lattice by X = X^{∗}(T), and the (abstract) weight lattice by P = Q^{⊥} (the dual being taken with respect to the Cartan pairing), let P^{+} denote the “cone” of dominant (abstract) weights, and let λ_{1}, ..., λ_{n} ∈ P^{+} denote the fundamental (abstract) weights. We have

Q ⊂ X ⊂ P.

Let λ denote a dominant character and let ρ_{λ} denote the irreducible representation whose highest weight is λ. Let X^{+} denote the “cone” of dominant characters. If ρ = ρ_{λ}, with λ ∈ X^{+}, let

χ_{ρ}(g) = χ_{λ}(g) = trace(ρ_{λ}(g)), g ∈ G.  (1)

Finally, let X^{θ} denote the subgroup of θ-invariant characters in X and let X^{θ+} denote the dominant ones. We call such characters θ-dominant.

It is clear that if λ ∈ X^{θ} then the restriction of λ to (1 − θ)T is trivial and hence λ is well-defined as a character of T_{θ}. Thus, there is a natural map η : X^{∗}(T)^{θ} → X^{∗}(T_{θ}). The kernel of the map t ⟼ t/θ(t), T → (1 − θ)T, is T^{θ}. Thus T/T^{θ} ≅ (1 − θ)T. If we knew that T^{θ} was connected (and hence T, T^{θ}, (1 − θ)T were all products of k^{×}’s) then we could conclude T_{θ} ≅ T^{θ}. This connectedness is proven in [11], §8. This also proves that each element t ∈ T can be uniquely factored t = t_{1}t_{2}, where t_{1} ∈ T^{θ} and t_{2} ∈ (1−θ)T (see also Corollaire 1.33 in [1]). From this fact, one can show that η is an isomorphism. These facts are summarized in the following lemma.

1As was pointed out by the referee, this condition is stronger than the notion of a quasicentral automorphism in the sense of [1]; Lemma 2.7 holds for quasi-central automorphisms but not Proposition 3.9 below.

2This is only a set-theoretic mapping since G_{θ−ss} is not in general a variety.

Lemma 3.1. Let G, θ, T be as in Lemma 2.6. There is a natural isomorphism

η : X^{∗}(T)^{θ} → X^{∗}(T_{θ}).

Furthermore, there is a (non-canonical) isomorphism T_{θ} ≅ T^{θ}. Indeed, each element t ∈ T can be uniquely factored t = t_{1}t_{2}, where t_{1} ∈ T^{θ} and t_{2} ∈ (1 − θ)T.

Proposition 3.2. Let ρ be a finite dimensional representation of G. The following are equivalent:

(a) ρ is θ-invariant,

(b) ρ extends to a finite dimensional representation of \overline{G}.

Remark 3.3. In the introduction, we stated that one aim of this paper is to provide a correspondence between irreducible representations of \overline{G} and irreducible representations of an endoscopic group. This is not exactly what we shall do. In fact, our correspondence will between θ-invariant representations of G (which extend to \overline{G}) and irreducible representations of an endoscopic group. Be that as it may, the remaining irreducible representations of \overline{G} can be provided with a similar correspondence (though with a different endoscopic group) thanks to the following well-known result.

Lemma 3.4. • If \overline{ρ} denotes an irreducible finite dimensional representation of \overline{G} then there is an irreducible representation ρ of G such that

– ρ^{θ^{m}} ≅ ρ, for some m|d and m is chosen as small as possible,

– ρ extends to the non-connected group G⋊ < θ^{m} >,

– \overline{ρ} ≅ Ind\overline{G} ; G⋊<θ^{m}>ρ.

• Let ρ be an irreducible finite dimensional representation of G and let θ be an automorphism of finite order d of G. If all d of the representations ρ^{θ^{i}}, 0 ≤ i ≤ d − 1, are inequivalent then Ind\overline{G} ; Gρ is an irreducible representation of \overline{G}.

Proof of the Lemma. This follows from a far more general result of Mackey [6] (see also Theorem 2, Ch III, §B of [5]).

Proof of the Proposition. Let (ρ, V_{ρ}) denote a finite dimensional representation of G. If ρ^{θ} ≅ ρ then there is an A ∈ GL(V_{ρ}) of finite order dividing d such that ρ(θ(g)) = A^{−1}ρ(g)A, for all g ∈ G. Define

\overline{ρ}(g.θ^{i}) = ρ(g)A^{−i}, g ∈ G, 0 ≤ i ≤ d.

where d denotes the order of θ. It may be verified that this defines an extension of ρ to \overline{G}, so (a) implies (b).

To prove the converse, (b) implies (a), let \overline{ρ} denote an extension of ρ to \overline{G}, so the image of \overline{G} under \overline{ρ} is also a semi-direct product. We may write

\overline{ρ}(g.θ^{i}) = ρ(g)A^{−i}, g ∈ G,

for some A ∈ GL(V). Calculating

ρ(gθ(g′))A^{−1} = \overline{ρ}(gθ(g′).θ) = \overline{ρ}(g.θ)ρ(g′) = ρ(g)A^{−1}ρ(g′) = ρ(g)ρ(θ(g′))A^{−1},

we find that ρ(g)ρ(θ(g′))A^{−1} = ρ(g)A^{−1}ρ(g′). This implies (a).

The following lemma is, as we shall see, an immediate consequence of the Weyl character formula.

Lemma 3.5. (a) If λ ∈ X^{θ+} then ρ_{λ} ≅ ρ_{λ}^{θ}. Conversely, if λ ∈ X^{+} and ρ_{λ} = ρ_{λ}^{θ} then λ ∈ X^{θ+}.

(b) Let λ^{θ} denote the highest weight of ρ_{λ}^{θ}, ρ_{λ^{θ}} = ρ_{λ}^{θ}. Let V = X ⊗ℝ, so we may extend the action of θ on X to V by linearity. These two actions (the above action of θ on V and the action of θ on dominant weights θ : λ ⟼ λ^{θ}) are compatible. For V as in the above lemma, let V^{θ} denote the subspace of θ-invariants

and, for any subset S ⊂ V , let S^{θ} = S ∩ V^{θ}.

Proof. (a) This follows from the direct construction of a representation with highest weight λ, as the referee pointed out. However, we give the following simple, analytic proof.

We have

χ_{λ} = \frac{A_{λ+ρ_{0}}}{A_{ρ_{0}}} ,  (2)

where ρ_{0} denotes half the sum of the positive roots of (B, T) and

A_{μ} = ∑_{w∈W}(−1)^{sgn(w)}wμ

is the alternating sum over the equivalence class [2], Theorem 24.2. Because of this, if λ ∈ X^{θ+} then

χ_{ρ_{λ}} = χ_{ρ_{λ^{θ}}} = χ_{ρ_{λ}^{θ}}.

This implies ρ_{λ} ≅ ρ_{λ}^{θ} since the character determines the equivalence class of the representation.

On the other hand, suppose ρ_{λ} ≅ ρ_{λ}^{θ}. The character formula above implies

\frac{A_{λ+ρ_{0}}(θ(t))}{A_{ρ_{0}}(θ(t))} = \frac{A_{λ+ρ_{0}}(t)}{A_{ρ_{0}}(t)},

for all t ∈ T. Since θ permutes the set of positive roots (it must since it preserves (B, T)), we have θ(ρ_{0}) = ρ_{0}. Thus,

\frac{A_{λ+ρ_{0}}(θ(t))}{A_{ρ_{0}}(θ(t))} = \frac{A_{θ(λ)+ρ_{0}}(t)}{A_{ρ_{0}}(t)},

which implies that ρ_{λ} = ρ_{λ}^{θ} has highest weight θ(λ). Since the highest weight is unique, it follows that λ = θ(λ) = λ^{θ}, as desired.

(b) Part (b) is a consequence of the above proof.

From this it follows that the irreducible, finite dimensional, θ-invariant representations are in 1-1 correspondence with the elements of X^{θ+}.

For w ∈ W and χ ∈ X, define (wχ)(t) = χ(w^{−1}tw) for all t ∈ T. Clearly, wχ ∈ X. If χ, χ′ ∈ X then we define

χ ∼ χ′

if there is an element of the Weyl group w ∈ W such that χ′ = wχ. This is an equivalence relation on X and the set of equivalence classes, i.e. W -orbits, will be denoted by X/W. The classes in X/W are in natural 1-1 correspondence with the set X^{+} since each class in X/W contains a unique dominant highest character.

If χ, χ′ ∈ X^{θ} then we define

χ ∼_{θ} χ′

if there is an element w ∈ W such that χ′ = wχ (we do not know if it suffices to assume w ∈ W^{θ} in this definition). This is an equivalence relation on X^{θ}.

Now define

Symm[χ] = ⊕_{χ′∼χ}χ′,

for any χ ∈ X. Note Symm[χ] ≅ Symm[χ′] if and only if χ ∼ χ′. Let

Symm_{θ}[χ] = ⊕_{χ′∼_{θ}χ}χ′,

for χ ∈ X^{θ}.

The following well-known lemma describes how the restriction of a irreducible character to a Cartan subgroup decomposes. We will prove the “twisted analog” of this result.

Lemma 3.6. (a) For λ ∈ X dominant,

χ_{λ}|_{T} = ∑_{μ ≤ λ μ dominant} m(μ)Symm[μ],

where the m(μ) ≥ 1 are integers satisfying m(λ) = 1. (When G is simply connected then all then m(μ) = 1.)

(b) For λ ∈ X dominant, there are ϵ_{μ} ∈ {±1}, for μ ≤ λ a dominant character, such that

Symm[λ] = ∑_{μ ≤ λ μ dominant} ϵ_{μ}m′(μ)χ_{μ}|_{T} ,

and ϵ_{λ} = 1. Here the m′(μ) ≥ 1 are integers satisfying m′(λ) = 1. (When G is simply connected then all then m′(μ) = 1.)

For a proof, see [10], §3.4.

Recall X^{θ+} ⊂ X denotes the subset of θ-invariant dominant characters. If ρ is θ-invariant, let \overline{ρ} denote an extension of ρ to \overline{G}. Write

\overline{ρ}(g.θ) = ρ(g)ρ(θ), g ∈ G,

for some ρ(θ) ∈ GL(V_{ρ}). This is an abuse of notation since the extension is not necessarily unique. ρ(θ) is only well-defined up to a d^{th} root of unity where θ^{d} = 1. We shall fix an extension in the following definitions. First, we claim that the trace of this endomorphism is, as a function of g, constant on the θ-conjugacy classes. Let

χ\overline{ρ}(g.θ^{i}) = trace(ρ(g)ρ(θ)^{i}), g ∈ G,

and, if λ ∈ X^{θ+} and ρ = ρ_{λ}, let

χ_{λ}^{θ}(g) = χ_{ρ}^{θ}(g) = χ\overline{ρ}(g.θ), g ∈ G.  (3)

This will be called the θ-character of ρ. From the fact that \overline{ρ}(g.θ) is a class function on G. < θ >, it follows that

χ_{ρ}^{θ}(y) = χ_{ρ}^{θ}(x^{−1}yθ(x)), x, y ∈ G,

if ρ is a θ-invariant finite dimensional representation of G.

Lemma 3.7. (a) For λ ∈ X^{θ+},

χ_{λ}^{θ}|_{T_{θ}} = ∑_{μ ≤ λ μ θ − dominant} m_{θ}(μ)Symm_{θ}[μ]|_{T_{θ}},

where the m_{θ}(μ) ≥ 1 are integers satisfying m_{θ}(λ) = 1.

(b) For λ ∈ X^{θ+}, there are ϵ_{μ} ∈ {±1}, for μ ≤ λ a θ-dominant character, such that

Symm_{θ}[λ]|_{T_{θ}} = ∑_{μ ≤ λ μ θ − dominant} ϵ_{μ}m′_{θ}(μ)χ_{μ}^{θ}|_{T_{θ}},

and ϵ_{λ} = 1. Here the m′_{θ}(μ) ≥ 1 are integers satisfying m′_{θ}(λ) = 1.

Remark 3.8. The “restriction” symbol |_{T_{θ}} used above is a slight abuse of notation (since T_{θ} is not a subset of T), which we hope the reader will pardon. Worst, it is not clear it is well-defined. It must be shown that both the lefthand side of (a) and the right-hand side of (b) above are well-defined. Let ρ denote an irreducible finite dimensional (complex) representation of G and let θ be a quasi-semisimple of G preserving a Borel pair (B, T). If ρ ≅ ρ^{θ} then, we claim, the restriction of its “twisted character” χ_{ρ}^{θ} to T_{θ} is well-defined. Indeed, if ρ is θ-invariant then the character χ\overline{ρ} is constant on conjugacy classes. Since χ\overline{ρ}(t′.θ) = χ\overline{ρ}(tt′θ(t)^{−1}.θ) = χ\overline{ρ}(t′tθ(t)^{−1}.θ), for all t, t′ ∈ T.

Proof. The proof of part (a) is analogous to Steinberg’s proof in the connected case ([10], §3.4). The restriction of χ_{λ}^{θ} to T_{θ} is a sum of characters (weights) μ in X^{∗}(T_{θ}), the multiplicity of each character occurring in the sum is the dimension of the corresponding weight space. If χ′ is any weight occurring in this decomposition then there is a w ∈ W and a dominant χ ∈ X^{∗}(T_{θ}) (regarded as an element in X^{∗}(T)^{θ} by Lemma 3.1) such that χ′ = wχ. Furthermore, these dimensions only depend on the W^{θ} -equivalence class of μ with respect to ∼_{θ} above. By Lemma 3.1, only θ-invariant weights can occur in this sum. The multiplicity with which (the “highest weight”) λ occurs in this sum is equal to the multiplicity of λ in \overline{ρ}_{λ}. But this is equal to the multiplicity of λ in ρ_{λ}, by construction (since they have the same representation space). Thus m_{θ}(λ) = 1, as desired.

Part (b) follows by inverting the (upper triangular) system of equations given in part (a).

Proposition 3.9. Let θ be as in Lemma 2.7. Assume that G contains no irreducible component of Cartan type A_{2n}. Let G^{1} denote the connected component of G^{θ} and let T^{1} denote the connected component of T^{θ}. The finite set R|_{T^{1}} forms a root system of the connected semisimple group G^{1} with maximal torus T^{1}.

Proof. This is a consequence of results in [9], §§12.16-12.19.

4. θ-conjugacy classes

Let G and \overline{G} be as in the previous section.

The following theorem, one of our main results, is an extension of a theorem of Steinberg to the non-connected case (see [10], §3.4, Theorem 2).

Theorem 4.1. Let G be a connected semi-simple group and let θ be as in Lemma 2.7.

(a) The restriction map

k[G.θ]^{G} → k[T.θ]^{N_{G}(T.θ)}

is an isomorphism.

(b) The “restriction” map

\begin{matrix}k[T.θ]^{N_{G}(T.θ)} → k[T_{θ}]^{W^{θ}} \\ f ⟼ res(f) \end{matrix}

is an isomorphism, where res(f)(t) = f(t.θ), t ∈ T_{θ} (see Remark 3.8).

(c) The functions {χ_{λ}^{θ}|_{T_{θ}} | λ ∈ X^{θ+}} form a basis for the k-vector space k[T_{θ}]^{W^{θ}}. The functions {χ_{λ} | λ ∈ X^{θ+}} form a basis for the k-vector space k[G.θ]^{G}.

Now we begin the proof of the theorem.

Proof. (a): Regarding the surjectivity of the restriction map, we need to know that if two elements of T.θ are G-conjugate then they are N_{G}(T.θ)-conjugate. This is a consequence of Lemma 2.7.

To prove injectivity, let f ∈ k[G.θ]^{G} be such that f|_{T.θ} = 0. If x ∈ G is θ-semisimple then there is a g ∈ G such that g^{−1}xθ(g) ∈ T by Lemma 2.7(a). Thus

f(x.θ) = f(g^{−1}xθ(g).θ) = 0,

since f is a class function. Recall x ∈ G is θ-semisimple if and only if x.θ ∈ \overline{G} is semisimple. Note also the image of the inclusion

\overline{G}_{ss} ↪ \overline{G}

is dense (almost all of the elements in \overline{G}, regarded as matrices, have distinct eigenvalues). It follows from these facts that f is zero on a dense subset. This implies that the restriction map is injective, which proves (a).

(b): We have N_{G}(T.θ)/T = W^{θ} by Lemma 2.6. Therefore,

k[T.θ]^{N_{G}(T.θ)} = k[T.θ]^{W^{θ}T} ≅ k[T_{θ}]^{W^{θ}}.

This proves (b).

(c): The first statement is a corollary of parts (a), (b), and the proof of Theorem 2(a) in §3.4 of [10]. The second statement follows from the first and parts (a), (b).

This completes the proof of the theorem.

5. Some corollaries

We list some corollaries of Theorem 4.1 above. These are all analogs of results of Steinberg in the non-connected case.

Let G and \overline{G} be as in the previous section.

Definition 5.1. We call G θ-simply connected (resp., θ-adjoint) and call \overline{G} simply connected (resp., adjoint) if X^{θ} = P^{θ} (resp., X^{θ} = Q^{θ}).

The following result can be proven by modifying the proof of Theorem 2, §3.4, in [10] and using the above proof.

Corollary 5.2. Assume G is θ-simply connected and let λ_{1}, ..., λ_{n} denote a set of θ-fundamental weights. Then {χ_{λ_{i}}|_{T_{θ}} | 1 ≤ i ≤ n} freely generates k[T_{θ}]^{W^{θ}} as a k-algebra.

Define k[G_{θ−ss}.θ] to be the vector space of functions on G.θ restricted to G_{θ−ss}.θ. The result below is a component of the proof of Theorem 4.1(a).

Corollary 5.3. The map

\begin{matrix}k[G.θ]^{G} → k[G_{θ−ss}.θ]^{G} \\ f ⟼ res(f) \end{matrix}

is an isomorphism of vector spaces, where res(f) denotes the restriction map.

Corollary 5.4. The θ-semisimple conjugacy classes of G are in 1-1 correspondence with the elements of T_{θ}/W^{θ}.

Remark 5.5. This Corollary is the twisted analog of Corollary 2, §3.4 in [10]. See also Remark 2.8 above.

Corollary 5.6. Let x = x_{0}.θ, y = y_{0}.θ ∈ G.θ be semisimple elements. The following are equivalent.

(a) x, y are G-conjugate,

(b) χ_{ρ}(\overline{x}_{0}) = χ_{ρ}(\overline{y}_{0}), for all ρ = ρ_{λ}, λ ∈ X^{θ+}. Here \overline{x}_{0}, \overline{y}_{0} ∈ T_{θ} denote the image of C_{θ}(x_{0}) ∩ T, C_{θ}(y_{0}) ∩ T in T_{θ} (which exists by Lemma 2.7).

Proof. ^{3} Follows immediately from the second statement in Theorem 4.1(c).

Corollary 5.7. If f ∈ k[G.θ]^{G} and x ∈ G then f(x.θ) = f(x_{θ−ss}.θ), for x ∈ G.

Proof. This follows from the second statement in Theorem 4.1(c) and the fact that χ_{λ}(x.θ) = χ_{λ}(x_{θ−ss}.θ), for all x ∈ G and λ ∈ X^{θ+}.

Corollary 5.8. A G-conjugacy class in G.θ is closed if it is semisimple.

Proof. We pick a basis of the vector space V , in the notation of §2 where \overline{G} ⊂ GL(V). We may identify each g ∈ \overline{G} with a matrix in GL(V).

For the proof, use the obvious twisted analog of the proof of Corollary 5 in §3.4, p. 92 of [10] (which relies on Corollary 5.6 above). In other words, fix a (semisimple) element x_{0}.θ ∈ G.θ. Let m_{0} denote the minimal polynomial of x_{0}.θ and let

S = {x.θ ∈ G.θ | χ_{λ}^{θ}(x_{0}) = χ_{λ}^{θ}(x), for all λ ∈ X^{+}^{θ}, and m_{0}(x.θ) = 0}.

S is Zariski closed and contains the conjugacy class of x_{0}.θ. Now let x.θ ∈ S. It is semisimple since its minimal polynomial has distinct roots (it divides m_{0}, which has distinct roots by definition). The hypothesis to Corollary 5.6 therefore holds and implies the statement of the corollary.

6. Character relations and endoscopic groups

Let θ be as in Lemma 2.7. As in Lemma 3.1, we fix an isomorphism

ψ : T^{θ} → T_{θ}.

The goal of this section is to show that, if G is a simple, simply connected (hence θ-simply connected) group and if θ is an automorphism of order d preserving a splitting (B, T, {X}) then there is a simple connected group H associated to G, θ such that 1. X^{θ+} is in natural 1-1 correspondence (defined in Lemma 3.1) with the

3I thank the referee for this short proof.

dominant characters of H, X_{H}^{+}, which we denote by

λ ↔ λ_{H},

2. there are distinct λ_{H,i} ∈ X_{H}^{+} such that we have a character relation

χ_{λ}^{θ}(A_{θ}(η(t).θ)) = ∑_{i=1}^{N} ζ_{i}χ_{λ_{H,i}}(t),  (4)

for all t ∈ T_{H}, where χ_{λ}^{θ} is as in (3) above, T_{H} is a regular torus, and A_{θ} is a map (to be defined below) from the semisimple conjugacy classes in G.θ to the semisimple conjugacy classes of H. Here the ζ_{i} ∈ ℂ (possibly all zero) depend on the extension of ρ_{λ} from G to G⋊ < θ > chosen in the definition of the θ-character.

As in [4], we have the following definition.

Definition 6.1. A group H as above is called a θ-endoscopic group.

Define H to be the group H = G^{θ} = {g ∈ G | θ(g) = g}. Since G is simply connected, H is connected, by Steinberg’s Theorem 8.1 in [11]. H has maximal torus T_{H} = T^{θ}, root lattice Q_{H}, character lattice X_{H} = X^{∗}(T_{H}), and weight lattice P_{H}, such that

Q_{H} = ℤ[R(B, T)|_{T^{θ}}] ⊂ X_{H} ⊂ P_{H} = Q_{H}^{⊥},

by Proposition 3.9. Furthermore, the Dynkin diagram of H is the “folded” Dynkin diagram of G (see [9], §12.18 and Theorem 12.19). We have T_{H} ≅ T_{θ}, W_{H} ≅ W^{θ} (see [11], §8). This implies H has property (1) of an endoscopic group. Furthermore, there is a 1-1 correspondence

A_{θ} : C_{G}(G_{θ−ss}.θ) → C_{H}(H_{ss}),  (5)

defined using the correspondences

C_{H}(H_{ss}) ↔ T_{H}/W_{H},
C_{G}(G_{θ−ss}.θ) ↔ T_{θ}/W^{θ},

and the non-canonical isomorphism

T_{H}/W_{H} ≅ T_{θ}/W^{θ}.  (6)

The following theorem is our other main result.

Theorem 6.2. Assume G and θ are as in Theorem 4.1 above. In addition, assume G is simply connected. Let H = G^{θ} be as defined above. There is an isomorphism of vector spaces

k[G.θ]^{G} → k[H]^{H},

induced by the restriction maps k[G.θ]^{G} → k[T_{θ}]^{W^{θ}}, k[H]^{H} → k[T_{H}]^{W_{H}}, and (6).

Proof. This follows from Theorem 4.1 and (6):

k[G.θ]^{G} → k[T.θ]^{N_{G}(T.θ)} → k[T_{θ}]^{W^{θ}}.

Remark 6.3. The equality claimed in (4) above, which is property (2) of an endoscopic group, is a special case of Lemma 3.7, using the definition of X_{H}^{+}.

7. Irreducible representations of \overline{G}, G simple

In this section, we use Dynkin diagrams to describe those irreducible finite dimensional representations of G which extend to representations of \overline{G}. This forms an important component of the complete description, which may be found in Remark 3.3 above. No proofs are given in this section. All statements are either proven in [2] or may be derived by modifying arguments there.

We remark that a relationship between the automorphisms of G and the automorphisms of the Dynkin diagram of G is given by Proposition 1.4.1 in [7].

G simply connected of type A_{n}

I thank the referee for pointing out that if n is even the restrictions of the roots in this case do not form a restricted root system in the sense of Proposition 3.9.

[figure: •; λ_{1}; •; λ_{2}; •; λ_{n−1}; •; λ_{n}]

Here

λ_{1}, the 1 − st fundamental weight ↔ the irreducible repn k^{n+1},

λ_{2}, the 2 − nd fundamental weight ↔ the irreducible repn ⋀^{2} k^{n+1},

and so on. The last node on the far right end is associated to the contragrediant of the standard representation:

λ_{n}, the n − th fundamental weight ↔ the irreducible repn ⋀^{n} k^{n+1}.

Let θ denote the automorphism θ(g) = J ^{t}g^{−1}J, where J denotes the skewdiagonal matrix whose skew-diagonal (top right to bottom left) is given by 1, −1, ..., (−1)^{n+1}. Then θ preserves the usual Borel pair (B, T) where B denotes the upper triangular subgroup and T the diagonal torus. Write the simple roots Δ = {α_{1}, ..., α_{n}} of (B, T) as usual, α_{1}(t) = t_{1}/t_{2}, ..., α_{n}(t) = t_{n}/t_{n+1}, where t = diag(t_{1}, ..., t_{n+1}). If we write the Dynkin diagram as usual, with α_{i} in place of λ_{i} above, then θ acts on the roots by

θ : α_{i} ↔ α_{n+1−i},

and on the weights by

θ : λ_{i} ↔ λ_{n+1−i},

for 1 ≤ i ≤ n. The θ-fundamental weights are then given by

\overline{λ}_{i} = \{ \begin{matrix}λ_{i} + θ(λ_{i}) = λ_{i} + λ_{n+1−i}, i < \frac{n+1}{2}, \\ λ_{i}, i = \frac{n+1}{2}, \end{matrix}

where 1 ≤ i ≤ \frac{n+1}{2} .

G simply connected of type D_{n}

[figure: •; λ_{1}; •; λ_{2}; •; λ_{n−2}; •; λ_{n−1}; •; λ_{n}]

λ_{1}, the 1 − st fundamental weight ↔ the irreducible repn k^{2n},

λ_{2}, the 2 − nd fundamental weight ↔ the irreducible repn ⋀^{2} k^{2n},

and so on. The third to the last node on the far right end is associated to the representation:

λ_{n−2}, the n − 1 − st fundamental weight ↔ the irreducible repn ⋀^{n−1} k^{2n}.

The top node on the far right is associated to the Spin^{+} representation:

λ_{n−1}, the n − th fundamental weight ↔ the irreducible repn Spin^{+}.

The bottom node on the far right is associated to the Spin^{−} representation:

λ_{n}, the n − th fundamental weight ↔ the irreducible repn Spin^{−}.

Assume n > 3. If n > 4 then the only non-trivial diagram automorphism is that which exchanges the two nodes on the far right end and leaves the others fixed. Let θ denote this automorphism (even when n = 4). In this case, the θ-fundamental weights are then given by

\overline{λ}_{i} = \{ \begin{matrix}λ_{i} + θ(λ_{i}) = λ_{i} + λ_{n+1−i}, i = n − 1, \\ λ_{i}, i < n − 1, \end{matrix}

where 1 ≤ i ≤ n − 1.

Now assume n = 4 and let θ denote the diagram automorphism such that

θ : α_{1} ⟼ α_{4} ⟼ α_{3}, θ(α_{2}) = α_{2}.

Using the fact that

λ_{1} = α_{1} + α_{2} + \frac{1}{2}α_{3} + \frac{1}{2}α_{4},
λ_{2} = α_{1} + 2α_{2} + α_{3} + α_{4},
λ_{3} = \frac{1}{2}α_{1} + α_{2} + α_{3} + \frac{1}{2}α_{4},
λ_{4} = \frac{1}{2}α_{1} + α_{2} + \frac{1}{2}α_{3} + α_{4},

we find that

θ : λ_{1} ⟼ λ_{4} ⟼ λ_{3}, θ(λ_{2}) = λ_{2}.

Therefore, the θ-fundamental weights are then given by

\overline{λ}_{i} = \{ \begin{matrix}λ_{1} + λ_{4} + λ_{3}, i = 1 \\ λ_{2}, i = 2, \end{matrix}

where 1 ≤ i ≤ 2.

G simply connected of type E_{6}

•λ_{1} •λ_{3} •λ_{4} •λ_{2} •λ_{5} •λ_{6}

Let θ denote an automorphism of G which acts on the roots by

θ : α_{1} ↔ α_{6}, α_{3} ↔ α_{5}, α_{4} ↔ α_{4}, α_{2} ↔ α_{2}.

Then from the table 1, [3], p. 69, we find that

θ : λ_{1} ↔ λ_{6}, λ_{3} ↔ λ_{5}, λ_{4} ↔ λ_{4}, λ_{2} ↔ λ_{2}.

The θ-fundamental weights are then given by

\overline{λ}_{i} = \{ \begin{matrix}λ_{1} + λ_{6}, i = 1, \\ λ_{3} + λ_{5}, i = 3, \\ λ_{2}, i = 2, \\ λ_{4}, i = 4, \end{matrix}

where 1 ≤ i ≤ 4.

Acknowledgments: This paper was written in 1995 and revised in 1998-99. I thank Will Traves and Roland Martin for many encouraging and useful conversations on this paper. Many thanks to both Prof. Kottwitz for helping with [4] and to the anonymous referee for correcting several errors in a previous version and generously providing many helpful suggestions.

References

[1] Digne, F., and J. Michel, Groupes reductifs non-connexes, Ann. Ecole Norm. Sup. 27 (1994), 345–406.

[2] Fulton, W., and J. Harris, “Representation theory, ” Springer Graduate Texts, 1994.

[3] Humphreys, J., “Introduction to Lie algebras and representation theory,” Springer Graduate Texts, 1972.

[4] Kottwitz, R., and D. Shelstad, “Foundations of twisted endoscopy, ” to appear in Asterisque.

[5] Lipsman, R.,, “Group representations: A survey of some current topics,” Springer Lecture Notes in Math. 388, 1974.

[6] Mackey, G., Unitary representations of group extensions, I, Acta Math. 99 (1958), 265–311.

[7] Satake, I., “Classification theory of semisimple algebraic groups,” Marcel Dekker, New York, 1971.

[8] Slodowy, P., “Simple singularities and simple algebraic groups,” Springer Lecture Notes in Math. 815, 1980.

[9]  Springer, T., “Linear algebraic groups,” Birkhäuser, Boston, 1983.

[10] Steinberg, R., “Conjugacy classes in algebraic groups,” Springer Lecture Notes in Math. 366, 1974.

[11] —, “Endomorphisms of linear algebraic groups, ” Memoirs of the Amer. Math. Soc. 80, 1968.

Mathematics Department U.S. Naval Academy Annapolis, MD 21402 wdj@usna.edu

Received January 17, 1999 and in final form February 23, 2000