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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.

5 votes
1 answer
1k views

kernel of G(Z/p^2 Z)->G(Z/pZ) is the lie algebra of G over Z/pZ?

Let G be an affine algebraic group defined over Z. The kernel of the natural homomorphism G(Z/p2Z)G(Z/pZ), if abelian, is a group which comes along w …
Amritanshu Prasad's user avatar
4 votes

What can representations of affine Weyl groups do?

Here is just one example (I know there are others too): Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are i …
Amritanshu Prasad's user avatar
6 votes

The product of non-commuting semisimple matrices need not be semisimple

(1000)(0110)=(0100)
Amritanshu Prasad's user avatar
4 votes

Infinite products of representations of the additive group

I address mainly Question C in the simplest special case where R is Q: In this case you are looking at locally nilpotent endomorphisms of a vector space. Similarity classes of such endomo …
Amritanshu Prasad's user avatar
19 votes

A slick proof of the Bruhat Decomposition for GL_n(k)?

My answer starts off just like Emerton's answer above; you want the G-orbits on G/B×G/B. But now, I diverge from Emerton to say that G/B is the space of full flags $F_0\subset F_1\subset \ …
Amritanshu Prasad's user avatar
10 votes
Accepted

Bruhat decomposition for G(R), R local ring or R=Z/p^r

Bruhat decomposition over Z/prZ is precisely the problem we looked at in this paper. We defined several invariants of double cosets, and classified the pairs (n,k) for which, when …
Amritanshu Prasad's user avatar
2 votes
Accepted

Gelfand pair and double coset decomposition

KπλK has a transitive right action of K. The stabilizer of Kπλ for this action is KπλKπλ. Thus, KπλK=xKπλx as x
Amritanshu Prasad's user avatar