All Questions
Tagged with rt.representation-theory reference-request
823 questions
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Representation theory over Z
In his answer to my question here, Victor Protsak quoted the following result:
Let $C_2$ be a finite cyclic group of order $2$. Then every $\mathbb{Z}[C_2]$ structure on $\mathbb{Z}^n$ is isomorphic ...
- 245
5
votes
1
answer
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Rallis inner product formula for U(2,2) and U(3)
Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are:
"A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998.
"An ...
- 467
4
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1
answer
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The geometry of closure of orbit of Borel subgroup in G/B × G/B.
Let $G$ be a reductive group, let $B$ be one of its Borel subgroups, and $T$ be a torus in $B$. $G/B$ is its flag variety. Let $y,w$ be two T-fixed points in $G/B$. Let $\mathcal{O}_{y,w}$ be the $B$-...
- 1,457
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2
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Reductive Lie algebra of a Lie group
In the answer of my question:
On the full reducibility of representations of reductive Lie algebras
James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in ...
- 671
7
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2
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780
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Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 52.9k
13
votes
3
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How to Compute the coordinate ring of flag variety?
Let $G$ be algebraic group over $\mathbb(C)$(semisimple), $B$ be Borel subgroup. consider flag variety $G/B$. It is known that $G/B$ is projective variety. So one consider the homogeneous coordinate ...
- 5,515
8
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3
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How to compute irreducible representation of Lie algebra in the framework of BBD
We know Beilinson-Bernstein established the following famous equivalence:
$D-mod_{G/B}\rightarrow U(g)-mod_{\lambda}$,where $G$ is algebraic group and $B$ is Borel subgroup, $G/B$ is flag variety of ...
- 5,515
9
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3
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Is there a good account of D-affinity and localization theorem for partial flag varieties?
Recall that a topological space is called $A$-affine for a sheaf of algebras $A$ if taking global sections of coherent sheaves of $A$-modules is an equivalence of categories to finitely generated $\...
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votes
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Several question on Affine Lie algebra
These questions might be elementary for I just started to learn affine Kac-Moody algebra.
It is well known that if we consider finite dimensional Lie algebra, we have the folloing projection:
$R(\...
- 5,515
2
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1
answer
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What is Extreme/Extremal vector according to some weights
I know this might be a very elementary question. But I could not find the original definition of Extreme(or Extremal)vectors of some weights $\lambda$(fixed the $w\in W$,where $W$ is Weyl group)
I am ...
- 5,515
14
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2
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How many ways are there to prove flag variety is a projective variety?
I am looking for references talking about different ways to prove flag variety $G/B$ is projective variety. Now I have some in mind:
There is a proof in Humphreys Linear algebraic groups, he first ...
- 5,515
5
votes
2
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346
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Reference request: A theorem by S. Garrison
A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). ...
- 2,508
12
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2
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Is Lusztig's conjecture solved?
What I said is Lusztig's conjecture about representation of quantum group at root of unity and representation of Lie algebra at positive characters.
It seems that Andersen-Jantzen-Soergel ever wrote ...
- 5,515
4
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2
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678
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What is the relationship between representations of Lie algebra and Weyl algebra?
Is there any paper talking about the relationship of representation of finite dimensional Lie algebra and Weyl algebra? Can we construct representations of Lie algebra from representations of Weyl ...
- 1,305
6
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2
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Does anyone have an electronic copy of Waldspurger's "Sur les coefficients de Fourier des formes modulaires de poids demi-entier"?
Is there an electronic copy of Waldspurger's paper "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" floating around the internet somewhere? This appeared in J. Pures Math. ...
- 13.1k
44
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The finite subgroups of SL(2,C)
Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
- 47.3k
6
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5
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3k
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Reference for quantum Schur-Weyl duality
I am trying to prove a version of quantum Schur-Weyl duality. I hope to be able to generalize the proof of the Schur-Weyl duality between $U_q(\mathfrak{gl}_n)$ and the Hecke algebra $H_r$. So I am ...
- 1,021
6
votes
2
answers
373
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Signed and unsigned Hecke algebra canonical basis
Consider the Hecke algebra $H_n$ of type $A_{n-1}$ with standard basis $T_w$, $w \in S_n$ with the quadratic relations $(T_s - u) (T_s + u^{-1}) = 0$ and braid relations. The unsigned canonical basis $...
- 1,021
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1
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Reference on a result on representation of moderate growth
Let G be a real reductive group, and P any parabolic subgroup. In the paper 'Canonical extensions of Harish-Chandra modules to representations of $G$' by Casselman, a result says that if we begin with ...
- 2,709
11
votes
1
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875
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An arithmetic highest weight theory?
I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps ...
- 436
17
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5
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3k
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Reference for this theorem in representation theory?
Let $G$ be a finite group and $\chi$ be an irreducible character of
$G$ (characteristic zero algebraically closed base field). If $H$ is
the kernel of $\chi$ then the irreducible representations of $G/...
- 887
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Complexes of representations with complementary central charges
This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
- 15.6k
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References for Lie superalgebras
Does anybody know good references to learn about Lie superalgebras? I started with Howe's "Remarks on classical invariant theory", which contains a study of osp(m,2n), and now I am reading Kac's '77 ...
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