**0**

votes

**0**answers

20 views

### When is the Ad (Adjoint Representation) Morphism a Closed Map

Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...

**3**

votes

**0**answers

97 views

### Restricting the Steinberg representation of $SL_{2n}$ over a finite field to the symplectic group

Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for $\text{SL}_n(\mathbb{F}_q)$.
What is the irreducible decomposition of the restriction of $\text{St}_{2n}(\mathbb{F}_q)$ ...

**1**

vote

**1**answer

107 views

### Tangent space of the Fourier algebra $A(G)$

Let $G$ be a real Lie group and $A(G)$ be its Fourier algebra. Let us call a linear continuous functional $f:A(G)\to{\mathbb C}$ a tangent vector of $A(G)$ in the point $a\in G$, if it satisfies the ...

**9**

votes

**0**answers

127 views

### Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

On the last page of Schmid's article "Discrete Series", he says
"In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure ...

**3**

votes

**0**answers

114 views

### What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...

**0**

votes

**0**answers

95 views

### Irreducible action of an algebraic group

Is the following claim true?:
Let $G$ be an algebraic group such that $G^\circ$ is reductive. Suppose $G$ acts irreducibly on $V$. Is it true that $V$ is decomposed (written as direct sum) into ...

**2**

votes

**1**answer

101 views

### When is the Fourier algebra $A(G)$ enough close to the Fourier-Stieltjes algebra $B(G)$?

I am reading P.Eymard's paper on the Fourier algebras of locally compact groups, and I have several questions about his constructions. I asked one of them in math.stackexchange, so far without ...

**1**

vote

**0**answers

46 views

### limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals (for example, in Gelbart-Jacquet, 1979):
$$\int_{Z_v N_v\backslash ...

**9**

votes

**2**answers

332 views

### on the center of a Lie group

I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles...
First some definitions:
• Let $G$ be compact, simple, and simply ...

**1**

vote

**0**answers

49 views

### Examples of Lie subalgebras of universal enveloping algebras

I'm looking for non-trivial examples of triples $(\mathfrak g, L, \psi)$, where $\mathfrak g$ and $L$ are finite-dimensional non-abelian Lie algebras over field $\Bbbk$ and $\psi\colon U_{\mathfrak ...

**5**

votes

**0**answers

78 views

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

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

**4**

votes

**1**answer

69 views

### Comodule analogue for statement that a faithful representation of an affine group scheme generates all

If $V$ is a faithful finite dimensional representation of an affine group scheme $G$ over a field $k$, then every finite dimensional representation of $G$ is isomorphic to a subquotient of $\otimes^n ...

**0**

votes

**1**answer

112 views

### Maximal subgroups of indefinite special orthogonal group SO(p,q)

Can someone answer the following question:
Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions:
...

**7**

votes

**1**answer

203 views

### What is the level of a positive energy loop group representation?

I am trying to learn a bit about loop group representation theory to understand its role in string geometry.
Let $G$ be a Lie group. I am thinking of $\text{Spin}(n)$, so you may assume $G$ to be ...

**3**

votes

**0**answers

117 views

### Symmetric power of an algebra

Given an algebra $A$ over $k$ with characteristic zero and a positive integer $n$, the subspace of $A^{\otimes n}$ consisting of all tensors invariant under the action of all permutations ...

**1**

vote

**1**answer

153 views

### Name for class of matrix determinants

Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $\alpha_i(x)$. Now, ...

**1**

vote

**2**answers

119 views

### When is the induced representation factored through the initial one?

Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ ...

**2**

votes

**0**answers

95 views

### quasi-split algebraic group [migrated]

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...

**5**

votes

**0**answers

363 views

### A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?

The following is my first question here on mathoverflow.
Let $M$ be a closed connected Riemannian manifold with an isometric effective action of a compact connected Lie group $G$. Consider the ...

**3**

votes

**0**answers

188 views

### Reference request: Beilinson-Bernstein for finite-dimensional reps and category O

I think I’ve once been told that under the Beilinson-Bernstein correspondence, finite-dimensional representations of a semisimple Lie algebra $\mathfrak{g}$ correspond to (twisted) D-modules on $G/B$ ...

**-1**

votes

**0**answers

68 views

### Confusion about the projected component in an irreducible space in the tensor product decomposition using Littlewood-Richardson?

The regular representation of the symmetric group can be formulated in terms of an abstract tensor, where the action of the symmetric group elements is by means of permuting the indices. Given an ...

**1**

vote

**0**answers

102 views

### Ext Quivers and their applications to Representation Theory

I am looking for references that provide an overview of the following two topics (it can be multiple references if necessary):
How to compute the Ext-quiver of a (locally finite or finite) ...

**2**

votes

**2**answers

169 views

### Jordan-Holder vs Harder-Narasimhan

Let $M$ be a module over an algebra or a group. I am interested the following decreasing filtration:
$F^0M=M$;
$F^iM$ is the smallest sub-module of $F^{i-1}M$ such that the quotient is ...

**2**

votes

**1**answer

154 views

### When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?

This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected.
It seems that the ...

**2**

votes

**0**answers

48 views

### Are Zariski-dense representations of a cocompact complex hyperbolic lattice non-obstructed?

Question
Suppose that $\Gamma < \text{SU}(n,1)$ is a cocompact lattice, and let $\rho \colon \Gamma \to G$ be a representation to a non-compact simple Lie group (most interesting case for me: $G = ...

**5**

votes

**1**answer

233 views

### Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc.
My question is ...

**5**

votes

**0**answers

187 views

### Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...

**9**

votes

**3**answers

288 views

### Maximal compact subgroup of p-adic lie groups

Let $k$ be a number field and $S$ be a finite set of places of $k$.
Let $G$ be a connected semisimple algebraic group over $k$.
Let $k_S=\prod_{v\in S}k_v$
where $k_v$ is the completion of $k$ at $v$. ...

**6**

votes

**1**answer

237 views

### An inequality on representations and subgroups of general linear groups over finite field

Let $q$ be a power of $p$, let $l$ be a prime different from $p$, and let $H_1$ and $H_2$ be two subgroups of $GL_n(\mathbb F_q)$ that are $l$-groups.
If for all characteristic $0$ representations ...

**13**

votes

**3**answers

643 views

### Moments of the trace of orthogonal matrices

Let $O_n$ be the (real) orthogonal group of $n$ by $n$ matrices.
I am interested in the following sequence which showed up in a calculation I was doing
$$a_k = \int_{O_n} (\text{Tr } X)^k dX$$
where ...

**3**

votes

**0**answers

56 views

### How well is the classification of low-dimensional semisimple Hopf superalgebras (or braided Hopf algebras) understood?

As far as I know, low-dimensional semisimple Hopf algebras are classified (along with non-semisimple ones) up to dimension 60, with the first example of a semisimple Hopf algebra not coming from a ...

**0**

votes

**0**answers

43 views

### Is the projector to irreducible tensor modules of SO(N) known?

To project a generic tensor to an irreducible module of SO(N) one has to (anti)symmetrize the indices and then subtract traces, e.g. for symmetric traceless 2-tensors
$$
\frac{1}{2} (\delta_{I_1 J_1} ...

**0**

votes

**0**answers

98 views

### Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every ...

**8**

votes

**2**answers

274 views

### Sums of degrees of irreducible complex characters

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to ...

**2**

votes

**1**answer

124 views

### Weight multiplicities for some particular representations of SO(2m).

I am looking for explicit formulas for the weight multiplicities of some particular irreducible representations of $SO(2m)$.
It is possible that they have been already computed; in this case I will ...

**2**

votes

**1**answer

62 views

### Pivotal functors of that are substantially different from finite group homomorphisms

Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a ...

**2**

votes

**0**answers

88 views

### Is every irreducible unitary class one representation induced?

Let $G$ be a connected semi simple Lie group with finite center.
Fix a maximal compact subgroup $K$.
An irreducible representation $(\pi,V)$ of $G$ is called a "class-one representation", if it ...

**2**

votes

**1**answer

170 views

### Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
...

**6**

votes

**0**answers

122 views

### minimal energy of affine Lie algebra reps

Let $\mathfrak g$ be a simple Lie algebra.
Let $\widetilde{L\mathfrak g}$ be the universal central extension of $L\mathfrak g:=\mathfrak g[t,t^{-1}]$. Let $V_\lambda$ be a positive energy ...

**-1**

votes

**0**answers

54 views

### Complex conjugation of positive roots [migrated]

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...

**2**

votes

**1**answer

189 views

### Irreducible representation of Heisenberg group with characteristic 2?

As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than ...

**1**

vote

**1**answer

106 views

### Representations of parabolic subgroups of the general linear group over the complex numbers

In all that follows, we are working over $\mathbb{C}$. Let $B \subseteq P \subseteq {\rm GL}(n)$ be a parabolic subgroup. Can you say anything in general about the representations of $P$? I suspect ...

**1**

vote

**0**answers

44 views

### “embedding” various matrix equivalences into the equivalence of particular linear map

Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" ...

**0**

votes

**1**answer

81 views

### Absolute irreducibility of a symmetric square?

This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so ...

**0**

votes

**0**answers

63 views

### reference help indecomposable representations of SL(2,R)

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible ...

**2**

votes

**1**answer

179 views

### What does the defect of a block measure?

In the context of decomposition matrices for Hecke algebras of finite Coxeter groups at a root of unity (such as the tables at the end of the book "Hecke algebras at a root of unity" by Geck-Jacon or ...

**4**

votes

**2**answers

120 views

### Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 ...

**0**

votes

**0**answers

174 views

### PBW proof proposal

One version of the PBW theorem states:
$\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras.
I am curious if this is a possible proof for the PBW theorem, part is taken ...

**3**

votes

**0**answers

105 views

### Non-linearly isomorphic non-equivalent $G-$representations?

Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow ...

**0**

votes

**0**answers

97 views

### Arthur's refinement of parameters for unitary automorphic representations

In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...