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Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

16 votes
Accepted

Simply connected simple algebraic groups

If $G$ is an adjoint algebraic group then it is always simple as an abstract group, (EDIT: This is because any proper normal subgroup of a simple algebraic group must be finite and lie in the centre). …
Jay Taylor's user avatar
  • 2,902
8 votes
Accepted

When the longest element of Weyl group is rational?

Let $B$ be a Borel subgroup containing $T$. As $F(B)$ and $B$ are both Borel subgroups containing $T$ there exists an element $n \in N_G(T)$ such that ${}^nF(B) = B$. Thus the Frobenius endomorphism $ …
Jay Taylor's user avatar
  • 2,902
7 votes
Accepted

The defining characteristic representations of Lie type groups

This result is false for a general connected reductive algebraic group $\mathbf{G}$ but is true if $\mathbf{G}$ is simple and simply connected. This was proved by Steinberg in Theorem 1.3 of the follo …
Jay Taylor's user avatar
  • 2,902
6 votes

Weyl group invariants in a maximal torus

Let us first assume that $G$ is a simple adjoint group and fix a maximal torus $T \leqslant G$. Let $W = N_G(T)/T$ be the corresponding Weyl group. What you're asking is, for a semisimple element $s \ …
Jay Taylor's user avatar
  • 2,902
5 votes
Accepted

Simple groups of Lie type

I'll just make my comment an answer so as to close this question. By Theorem 24.17 of the book "Linear Algebraic Groups and Finite Groups of Lie Type" by Malle and Testerman we have the answer is yes …
Jay Taylor's user avatar
  • 2,902
5 votes
Accepted

Finite field analogue of representations in same packet have equal central character

This is quite an old question but I believe the answer to your question is given in Lemma 2.2 of Malle's paper "Height 0 characters of finite groups of Lie type" (2007) which is freely available onlin …
Jay Taylor's user avatar
  • 2,902
5 votes
Accepted

Centralizers of $\mathbb{F}_q$-rational semisimple elements of a finite group of Lie type

As @LSpice already pointed out, you need $q$ to be sufficiently large even in the case of a Levi subgroup. Just take $G = \operatorname{GL}_n(\overline{\mathbb{F}}_q)$ and $G^F = \operatorname{GL}_n(\ …
Jay Taylor's user avatar
  • 2,902
3 votes
Accepted

A bijection between Lusztig series induced by inflation

Your approach is correct and is proven in the book by Digne-Michel (in fact a more general statement is proven there). Indeed, by Proposition 13.22 in Digne-Michel we know that $$R_{T\subseteq B}^G(\ …
Jay Taylor's user avatar
  • 2,902
3 votes
Accepted

Regular embeddings of reductive groups

I had cause to think about this exercise recently so I thought I’d write an answer. I think Jim’s answer is sufficient but as you seem to want more details I’ll provide them here. I am aware that you …
Jay Taylor's user avatar
  • 2,902
3 votes

The centralizer of a semisimple element which is not contained in any proper parabolic subgr...

What you want is a specific part of something that fits into a wider framework due to Borel--Tits. Specifically, you're looking for Theorem 4.15 and Corollary 4.16 of "Groupes réductifs", Inst. Hautes …
Jay Taylor's user avatar
  • 2,902
3 votes
1 answer
1k views

Richardson Classes and the Bala Carter Theorem

I am interested in trying to understand the following problem. Let $G$ be a connected simple algebraic group of type $D_n$, (with $n\geqslant 4$ even), defined over an algebraically closed field of od …
Jay Taylor's user avatar
  • 2,902
2 votes
0 answers
171 views

Springer Isomorphisms for Adjoint Simple Exceptional Groups

I'm trying to understand explicitly a construction of Springer isomorphisms for adjoint exceptional groups given by Bardsley and Richardson. Their construction is as follows. Let $G$ be an adjoint sim …
Jay Taylor's user avatar
  • 2,902
1 vote

Regular elements in the torus of a group of Lie type

This is an old question now but I had cause to look at it recently. I thought it was worthwhile pointing out that Carter's proof about the existence of nondegenerate maximal tori in Proposition 3.6.6 …
Jay Taylor's user avatar
  • 2,902
1 vote

Examples of non-split algebraic groups

If you allow yourself to work over fields of positive characteristic then finite reductive groups give you a raft of examples for (1). Indeed, let $G$ be a connected reductive algebraic group over $\m …
1 vote
1 answer
570 views

Decomposing Semisimple Perverse Sheaves

So I asked this on maths SE because I don't truly consider it to be a research level question. This question mostly arises out of my completely limited understanding of perverse sheaves. However I do …
Jay Taylor's user avatar
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