**5**

votes

**2**answers

597 views

### Is there a topological Chevalley-Shephard-Todd Theorem?

Is the following true:
For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections.
( ...

**3**

votes

**0**answers

822 views

### 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. ...

**5**

votes

**1**answer

189 views

### 1-dimensional representations of the affine Hecke algebra for $SL_2$

Kazhdan-Lusztig theory gives a correspondence between irreducibles of the affine Hecke algebra for a simply connected linear algebraic group $G$ and certain homological data extracted from the ...

**8**

votes

**2**answers

626 views

+300

### A question on representation of graphs

Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ ...

**3**

votes

**3**answers

328 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

**4**

votes

**1**answer

82 views

### Characterizations of Jacobson-Morozov parabolics associated to a nilpotent

Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic ...

**22**

votes

**2**answers

953 views

### Majorization and Schur Polynomials

Let me first define the majorization order (or dominance order) on partitions as $\lambda \succeq \mu$ iff $$\sum _{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$$ for all $1\le k\le l-1$ and ...

**4**

votes

**2**answers

165 views

### What are the “tensor-closed” object of the BGG category $\mathcal{O}$ of a semisimple Lie algebra $\mathfrak{g}$?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra and we can consider its BGG category $\mathcal{O}$. It is well-known that $\mathcal{O}$ is not closed under tensor product, ...

**0**

votes

**0**answers

44 views

### Stabilizer subgroup in adjoint action [migrated]

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...

**15**

votes

**6**answers

2k views

### Faithful representations and tensor powers

The following result was mentionned earlier in this thread, I searched a bit in the related threads and couldn't find a proof. I would really like to see a proof of it:
Let $G$ be a finite group and ...

**5**

votes

**0**answers

156 views

### Examples of Rankin-Selberg L-functions from Eisenstein series

I've been digging for awhile to not much success, so I figure I would try here:
I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...

**4**

votes

**1**answer

168 views

### highest weight the half-sum of positive roots

Sorry if this one is already asked - couldnt find anything about it.
If I take the irreducible representation of $GL_n$ whose highest weight is the half-sum $\rho$ of positive roots, it has ...

**5**

votes

**1**answer

129 views

### Do Iwahori-Hecke algebras come from cohomology classes?

Let $W$ be a Coxeter group. The Iwahori-Hecke algebra $H_q(W)$ is a deformation of $k W$.
Question: is there some way to interpret the deformation $H_q(W)$ as a cohomology class? It doesn't ...

**0**

votes

**1**answer

203 views

### Determinants of tensors [closed]

Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like ...

**32**

votes

**2**answers

3k views

### Open problems/questions in representation theory and around?

What are open problems in representation theory?
What are the sources (books/papers/sites) discussing this?
Any kinds of problems/questions are welcome - big/small, vague/concrete.
Some estimation ...

**3**

votes

**1**answer

162 views

### A representation of a finite group where every nonzero vector has a trivial stabilizer [duplicate]

What are the finite groups which admit a non-zero representation in char 0 where every non-zero vector has stabilizer equal to $\left<1\right>$? Cyclic groups of prime order is one obvious ...

**3**

votes

**1**answer

196 views

### reference request: direct product of WOT-continuous unitary representations

In an article I'm revising, I spend some time giving a self-contained proof of the following result
Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary ...

**2**

votes

**0**answers

280 views

### When a Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical.
My question is When a Spherical variety is $K$-stable? Is ...

**3**

votes

**1**answer

166 views

### An equivalence of derived categories by Happel-Reiten-Smalø

I have a problem in understanding the proof of a theorem by Happel-Reiten-Smalø. The original reference is this article
http://arxiv.org/abs/0911.4473
.
I write down the text of the theorem and a ...

**0**

votes

**1**answer

207 views

### Are such averages known with representations of $S_n$?

Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to ...

**0**

votes

**1**answer

93 views

### Mysterious central projections in the full group $C^*$-algebra

Let me quote the following theorem about the structure of $C^*(G)$ for property $T$ group (the reference is Higson and Roe "Analitycal K-homology"):
Let $G$ be a property $T$ (discrete) ...

**6**

votes

**2**answers

197 views

### regular semisimple elements on spherical varieties

Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$).
What can ...

**0**

votes

**0**answers

50 views

### Quadratic Casimirs of the E7 series

Of course a) you can compute the QCs for any member and irrep of the $E_7$ series ($E_7,H_{32},D_6$..., corresponding to $m=8,6,4$...) by standard liealgebraology and b) in general form via the Vogel ...

**12**

votes

**1**answer

796 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

**5**

votes

**1**answer

262 views

### Representations of the unit group in a ring of integers

Let $K/\mathbb{Q}$ be a finite extension of degree $d > 1$. Suppose that $\omega_1, \cdots, \omega_d$ is a basis for $K$ over $\mathbb{Q}$. Further, we assume that $\omega_1, \cdots, \omega_d \in ...

**8**

votes

**5**answers

711 views

### Applications of Chevalley Restriction Theorem

Let $G$ be a simple linear algebraic group (over $\mathbb{C}$, say) and $\mathfrak{g}$ be its Lie algebra, $\mathfrak{t}\subset \mathfrak{g}$ the Lie algebra of a maximal torus in $G$ and $W$ the ...

**1**

vote

**1**answer

45 views

### Decomposition of quadratic polynomials inti irreducible representations of affine group over a finit field

Let $\mathbb{F}_p$ be a finite field of order $p$ and $G$ be the general affine group of degree one over this finite field. Further let $V$ denote the quadratic polynomials over $\mathbb{F}_p$. I ...

**3**

votes

**0**answers

56 views

### Homological dimension of Joseph quotients

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ not isomorphic to $sl(n)$.
Let $\mathcal O$ be the minimal nilpotent orbit in $\mathfrak g^*$. Joseph proved that there exists unique ...

**5**

votes

**0**answers

59 views

### Ring of SO(n)-invariant differential operators on M_n,m

I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).
There comes a point in the paper (Lemma 2.8) ...

**10**

votes

**1**answer

2k views

### Is the Duflo polynomial conjecture open?

Let $G/K$ be a symmetric space. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition,
with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant
...

**-1**

votes

**0**answers

32 views

### General form of a matrix $M$ commutes with the unitary representation $U^{\otimes m},~ \forall U\in U(n)$ [migrated]

My question is about the general form of a $n^m\times n^m$ positive definite matrix $M$ where
$$[M,U^{\otimes m}]=0,~ \forall U\in U(n)$$
or in other words, M commutes with all members of the the ...

**4**

votes

**0**answers

428 views

### Is a non-trivial finite perfect group of order 4n? [migrated]

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial.
Question: Is a non-trivial finite perfect group of ...

**2**

votes

**0**answers

88 views

### Reference request: proofs of the theorems in the paper On the representation of the group GL(n, K) where K is a local field

In the paper On the representation of the group GL(n, K) where K is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers. I ...

**0**

votes

**1**answer

73 views

### Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$

I have the following questions:
Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra ...

**0**

votes

**0**answers

97 views

### Unitarizability of group representations

Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...

**2**

votes

**2**answers

301 views

### $t$-analogue of the symmetric power of an additive character over $\Bbb{F}_q^*$

Let $G$ be a finite group and let $f: G \longrightarrow \Bbb{C}$ be any complex-valued function. For integers $k, n \geq 0$, an indeterminant $t$, and $x \in G$ let $f_k(x) := f \big( x^k \big)$ and ...

**5**

votes

**0**answers

102 views

### Fundamental lemma: why is the transfer factor a power of q

Let $k$ be a finite field of sufficiently large characteristic, $F = k((t))$ and $\mathfrak{o} = k[[t]]$. Let $G$ be a reductive algebraic group defined over $\mathfrak{o}$. Roughly stated, for sake ...

**4**

votes

**1**answer

130 views

### Infinite-dimensional admissible representations of SL(2,C)

I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...

**14**

votes

**4**answers

3k views

### Cohomology of Flag Varieties

For $K$ a compact Lie-group with maximal torus $T$, I'd like to know the cohomology $\text{H}^{\ast}(K/T)$ of the flag variety $K/T$.
If I'm not mistaken, this should be isomorphic to the algebra of ...

**2**

votes

**0**answers

73 views

### Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block.
Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = ...

**2**

votes

**1**answer

94 views

### Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group

Recently in a paper we get the following result:
Let a discrete group $\Gamma$ act on a discrete abelian group $G$ by group automorphisms. Every irreducible unitary representation $\pi$ of ...

**2**

votes

**1**answer

327 views

### Decomposing a reducible representation of the unitary group

Consider the representation $L_U$ of the unitary group $U(n)$ on $L(\mathbb{C}^n)$ where $L_U$: $L(\mathbb{C}^n) \rightarrow L(\mathbb{C}^n)$ is a linear operator that $L_U M=U M U^{\dagger} $, ...

**1**

vote

**0**answers

108 views

### The representation theory for the fake Heisenberg groups over non-perfect local field

Let $K$ be a local field of characteristic $p$, where $p$ is a prime number greater than 2. In particular, $(x+y)^p=x^p+y^p$ for $x,y\in K$.
The fake Heisenberg group is defined to be
$$
...

**5**

votes

**1**answer

166 views

### What is Jantzen's formula for the determinant of the Shapovalov form in the case of generalized Verma modules?

The best reference I found is
[Kac, Kazhdan '79]
which extends the results of Shapovalov and Jantzen to the case of infinite dimensional Lie algebras.
Theorem 1 of this paper gives the Shapovalov ...

**10**

votes

**1**answer

523 views

### $q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?

Define n-quantum vector space to be the algebra
$$
{\mathbb C}_q^n := \mathbb{C}\left< x_i \mid i =1, \ldots, N\right>/\left<x_i x_j = q x_j x_i \mid i<j\right>.
$$
For $q=1$, we get ...

**-1**

votes

**0**answers

105 views

### Compact finite dimensional group

Suppose that $G$ is a compact, finite dimensional topological group (finite dimensional as a topological space). Does it follows that $G$ can be faithfully represented on some $U(n)$ (in other words, ...

**20**

votes

**7**answers

1k views

### Computer package for representation theory of the symmetric group

Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S_7$):
(1) $V \otimes W$
(2) $S_\lambda V$, where $S_\lambda$ is a ...

**3**

votes

**1**answer

636 views

### Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...

**4**

votes

**1**answer

401 views

### A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...

**3**

votes

**1**answer

149 views

### Is every weight of an integrable highest weight module in the Tits cone?

Let $\mathfrak{g}$ be a Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, Weyl group $W$, and simple roots and coroots $\alpha_i, \check{\alpha_i}, i \in I$, respectively. Let $L$ be an ...