Automorphisms and Generalized Involution Models of Finite Complex Reflection Groups

On the affine Schützenberger involution

We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a certain Green polynomial of type $A$ evaluated at -1.

Coxeter group actions on the complement of hyperplanes and special involutions

We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on the total cohomology H(M_G,C) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups I_2 (2k+1) and that the standard action has no a…

Conjugacy Class Properties of the Extension of GL(n,q) Generated by the Inverse Tranpose Involution

Letting tau denote the inverse transpose automorphism of GL(n,q), a formula is obtained for the number of g in GL(n,q) so that gg^{tau} is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg^{tau} behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n,q) and of symmetric fun…

Generalizing the MVW involution, and the contragredient

For certain quasi-split reductive groups $G$ over a general field $F$, we construct an automorphism $ι_G$ of $G$ over $F$, well-defined as an element of ${\rm Aut}(G)(F)/jG(F)$ where $j:G(F) \rightarrow {\rm Aut}(G)(F)$ is the inner-conjugation action of $G(F)$ on $G$. The automorphism $ι_G$ generalizes (although only for quasi-split groups) an involution due to Moeglin-Vigneras-Waldspurger in [MVW] for classical groups which takes any irreducible admissible representation $π$ of $G(F)$ for $G$ a classical group and $F$ a local field, to its contragredient $π^\vee$. The paper also formulates a…

On Kottwitz' conjecture for twisted involutions

Kottwitz' conjecture is concerned with the intersections of Kazhdan--Lusztig cells with conjugacy classes of involutions in finite Coxeter groups. In joint work with Bonnafé, we have recently found a way to prove this conjecture for groups of type $B_n$ and $D_n$. The argument for type $D_n$ relies on two ingredients which were used there without proof: (1) a strengthened version of the "branching rule" and (2) the consideration of "$\diamond$-twisted" involutions where $\diamond$ is a graph automorphism. In this paper we deal with (1), (2) and complete the argument for type $D_n$; moreover, w…

Semilinear automorphisms of classical groups and quivers

For a classical group $G$ over a field $F$ together with a finite-order automorphism $θ$ that acts compatibly on $F$, we describe the fixed point subgroup of $θ$ on $G$ and the eigenspaces of $θ$ on the Lie algebra $\mathfrak{g}$ in terms of cyclic quivers with involution. More precise classification is given when $\mathfrak{g}$ is a loop Lie algebra, i.e., when $F=\mathbb{C}((t))$.

MVW 对合（metaplectic 群）

The MVW involution of the metaplectic group

本文证明了非 Archimedean 局部域上 metaplectic 群存在 MVW 对合，推广了一般线性群的转置-逆自同构，适用于特征异于 $p$ 的系数域。

On involutions in symmetric groups and a conjecture of Lusztig

Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions was naturally endowed with a module structure of the Hecke algebra of $(W, S)$. Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter $u^2$) associated to $(W,S)$ generated by the element $X_{\emptyset}:=\sum_{w^\ast=w}u^{-\ell(w)}T_w$. In this paper we prove this conjecture in the case when $\ast=\text{id}$ …

Frobenius--Schur indicators of unipotent characters and the twisted involution module

Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphism of $W$. We show a remarkable relation between the $\sg$-twisted involution module for $W$ and the Frobenius--Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig-Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius--Schur indicators for "unipotent characters" of twisted dihedral groups.

Signs, involutions and Jacquet modules

Let $G$ be a connected reductive $p$-adic group and let $θ$ be an automorphism of $G$ of order at most two. Suppose $π$ is an irreducible smooth representation of $G$ that is taken to its dual by $θ$. The space $V$ of $π$ then carries a non-zero bilinear form $(\mspace{7mu},\mspace{6mu})$, unique up to scaling, with the invariance property $(π(g)v, π({}^θg)w) = (v,w)$, for $g \in G$ and $v, w \in V$. The form is easily seen to be symmetric or skew-symmetric and we set $\varepsilon_θ(π) = \pm1$ accordingly. We use Cassleman's pairing (in commonly observed circumstances) to express $\varepsilon_…

Involutory reflection groups and their models

A finite subgroup of $GL(n,\mathbb C)$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups.

Non-commutative groups as prescribed polytopal symmetries

We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.

Unitary and symmetric units of a commutative group algebra

Let F be the field of two elements and G a finite abelian 2-group with an involutory automorphism. The extension of this automorphism to the group algebra FG is called an involutory involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.

Monoidality of Kato's Reflection Functors

Kato has constructed reflection functors for KLR algebras which categorify the braid group action on a quantum group by algebra automorphisms. We prove that these reflection functors are monoidal.

量子环面代数的自同构：来自扩展双仿射辫群的作用

Automorphisms of quantum toroidal algebras from an action of the extended double affine braid group

本文构造扩展双仿射辫群对量子环面代数的作用，并利用其产生交换水平和垂直子代数的自同构，推广了 Miki 在 A 型中的结果。