THE GROUP GLN(K) o Z=2Z 43
Comparing the two ends of this equation, we see that 1 and 2 must lie in the same connected component of AutG.
If they are both inner automorphisms, then they must define isomorphic semi-direct product. In fact, the resulting semi-direct product is the direct product. To see this, let 2 be the trivial automorphism. We can always compose with an automorphism of G (extending to G <2>) such that jG = Id, then the above equation reads 1(g) = xgx1 for some x satisfying x2 = 1. That is, 1 = ad x for x2 = 1, but these are exactly the inner involutions.
If 1 and 2 are both outer automorphisms, then we need some explicit information about the group G. But let us first note that if 1 and 2 are G-conjugate, then they define isomorphic semi-direct product. Suppose 1 = and 2 = yy1 for some outer involution and y 2 G. Then we put x = (y)y1. Again, assume jG = Id, and so (x2)2 = 1 and 1(g) = x2(g)x1 for all g 2 G, as required.
We now restrict ourselves to G = GLn(k). We have already said that GLn(k) has two distinct conjugacy classes of outer involutions when n is even. Suppose 1 is of symplectic type and 2 is of orthogonal type and that there is an isomorphism between the two semidirect products. Then in the group Go<2>, 22 = 1 by definition, and (x2)2 = 1 as the image of 21
. But x2 is of symplectic type as its action on G is the same as 1, assuming jG = Id. We deduce that, modulo k, x2 is conjugate to t2, where t is as in (II.22.214.171.124). Since (t2)2 = 1, and for any z 2 k (zt2)2 = (t2)2, we have (x2)2 = 1, which is a contradiction.
We will write s¯G or o¯G to indicate whether 0 = s or o, and write ¯G when there is no need to distinguish them. Note however, that the above classification also works for SLn(k). But in PGLn(k), since there is no dierence between 1 and 1, the two isomorphism classes degenerate, and they are actually isomorphic to Aut(G). If no confusion arises, we can also denote by ¯G the direct product G Z=2Z. By definition, is a quasi-central element in G:.
Irreducible Subgroups of GLn o<>
II.5.4.1 Maximal Parabolic Subgroups of G o Z=2Z We are only interested in those parabolic subgroups that meet both connected components of ¯G = GoZ=2Z. If GoZ=2Z G Z=2Z, then a maximal parabolic subgroup is just the union of two copies of a maximal parabolic subgroup of G, one copy in each connected component.
Now let G o Z=2Z be defined by some graph automorphism . For G = GLn(k), there will be no dierence between s¯G and o¯G for the present problem, so we will not specify the conjugacy class of 0. We conclude from §II.1.1.5 that if T is the maximal torus of the diagonal matrices and B is the Borel subgroup of the upper triangular matrices, then every maximal standard parabolic subgroup P of G containing B such that N¯G(P) meets G:0 is of the following .
We list below some basic properties.
(i) If (˜; h) is a (; )-invariant pair, then (x˜; ( (x)hx1)) is also a (; )-invariant pair.
(ii) If (˜; h) is a (; )-invariant pair, then (˜; hz) is also (; )-invariant for any z 2 StabG(˜).
(iii) Let (˜; h) be a (; )-invariant pair. Let 0 : ! AutG be another homomorphism.
Suppose for all 2 , 0 1 = ad x, for some x 2 G. Then ˜ can be completed into a (; 0)-invariant pair by defining h0 = xh. (iv) If (˜; h) is a (; )-invariant pair for a choice of , then for another 0 , the pair (˜; ( (˜(1 ))h)) is (; )-invariant, with = 1 0 . In the case of (iii), we say that and 0 are similar. So if and 0 are similar, then ˜ is (; )-invariant if and only if it is (; 0)-invariant. Similarity classes are parametrised by the set of homomorphisms of discrete groups Hom(;A(G)).
III.3.2.1 Let P be a proper parabolic subgroup of G and let E be a principal G-bundle. Recall that a reduction of E to P is a principal P-bundle P and an isomorphism E P P G, where P acts on G by left multiplication.
For 2 , the G-conjugacy class of P is -stable if (P) is G-conjugate to P. If the Gconjugacy class of P is -stable for all 2 , then we say that it is -stable. The G-conjugacy class of P is -stable if and only if N¯G(P) meets G:s1 . In this case, denote by P the connected component of N¯G (P) contained in G:s1 . If P is a principal P-bundle for some -stable P, then we can define P as P P P.
Table of contents :
I.1 Character Varieties
I.2 Character Table of GLn(q) o<>
II Preliminaries on Representations and Algebraic Groups
II.1 Notations and Generalities
II.2 Finite Classical Groups
II.3 Non-Connected Algebraic Groups
II.4 Generalised Deligne-Lusztig Induction
II.5 The Group GLn(k) o Z=2Z
III Character Varieties with Non-Connected Structure Groups
III.1 G o -Character Varieties
III.2 Irreducibility and Semi-Simplicity
III.3 Flat Connections
III.4 Monodromy on Riemann Surfaces
III.5 Double Coverings
IV The Character Table of GLn(q) o<>
IV.1 Parametrisation of Characters
IV.2 Parametrisation of Conjugacy Classes
IV.3 Shintani Descent
IV.4 Character Sheaves
IV.5 Extensions of -Stable Characters
IV.6 The Formula
V E-polynomial of GLn(C) o<>-Character Varieties
V.1 The Point-Counting Formula
V.2 Symmetric Functions Associated to Wreath Products
V.3 Miscellany of Combinatorics
V.4 Computation of the E-Polynomial
A Character Tables in Low Ranks