**3**

votes

**0**answers

45 views

### Some questions on analytic vectors and the integrability of Lie-algebra representations

I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...

**1**

vote

**1**answer

51 views

### Does restriction to an open subgroup preserve projective smooth representations?

Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth ...

**2**

votes

**1**answer

90 views

### Compact induction as a tensor product

Let $G$ be a locally profinite (i.e., locally compact Hausdorff and totally disconnected) topological group, $H \le G$ a closed subgroup, and $(W, \sigma)$ a representation of $H$ over $\mathbb{C}$ ...

**2**

votes

**0**answers

64 views

### Lattice model for Affine Grassmannians of non type A

There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional ...

**3**

votes

**0**answers

64 views

### Annihillator of the highest weight vector in a finite-dimensional representation

Let $\mathfrak g$ be a simple complex Lie algebra and let $V(\lambda)$ be a finite-dimensional representation with highest weight $\lambda$. Let $v$ be the highest weight vector. Then the action of ...

**3**

votes

**2**answers

263 views

### Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series
$$
E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}},
$$
where $z=x+iy$. We think of $E(z,s)$ as a ...

**0**

votes

**1**answer

57 views

### Nagakami behavoir

Is the sum of square Nagakami random variables Erlang distributed?
What is the distribution of euclidean norm of complex Nagakami?
Cheers!

**1**

vote

**1**answer

122 views

### Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$

Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory.
I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...

**2**

votes

**1**answer

122 views

### Various definitions of the Bruhat decomposition of the affine Grassmannian

Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ ...

**0**

votes

**0**answers

52 views

### Generalized weight space

In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space:
If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function ...

**1**

vote

**0**answers

33 views

### Classification properties of fusion rings

Fusion rings have so many classification properties (I checked the literature a bit) that my head hurts. For practical reasons I define the following three new properties (which might coincide with ...

**1**

vote

**0**answers

31 views

### Anti-Invariant Polynomials of the Dihedral group

I'm interested in the one-dimensional irreducible representations of $D_{2n}$ acting on $\mathbb{R}[x,y]$. I have found that the trivial representations for an algebra freely generated by $x^2+y^2$ ...

**2**

votes

**0**answers

92 views

### Invariant Theory over finite adeles

Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...

**1**

vote

**1**answer

120 views

### Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket
$$
[e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f.
$$
For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...

**2**

votes

**2**answers

131 views

### Projection formula for smooth representations of locally profinite groups

Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ ...

**1**

vote

**0**answers

62 views

### Fundamental invariants for root subsystems

Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$.
Now suppose $\Psi$ is ...

**3**

votes

**4**answers

332 views

### Representations of the two dimensional non-abelian Lie algebra

I would like to know a complete description of the indecomposable representations of the two dimensional non-abelian Lie algebra over the complex numbers. The finite dimensional representations would ...

**4**

votes

**1**answer

140 views

### Shift-invariant symmetric functions in representation theory?

The connection between symmetric functions and representation theory is well-known.
Now consider the subspace of symmetric functions that are shift-invariant,
that is, functions satisfying ...

**7**

votes

**1**answer

166 views

### On unramified p-adic groups

Let G be a reductive group over a local field F. Let O be the ring of integers of F.
The following are equivalent (and groups satisfying these conditions are called unramified):
(a) G is quasisplit ...

**2**

votes

**0**answers

90 views

### Applying Lemma 2.12 of Deligne's/ Milne's “Tannakian Categories” on an irreducible representation

since this is my first question here, I'm not very certain whether I state it properly. I'm thankful for any helpful remarks.
Currently I'm trying to understand the above-mentioned article which can ...

**2**

votes

**1**answer

74 views

### Weight polytopes of the fundamental representations of simple Lie groups

Where can I find a description of the weight polytopes of the fundamental representations of the classical complex simple Lie groups?
Thanks in advance

**4**

votes

**0**answers

170 views

### Decompositions of a compact Lie group into “fixed point set types”

Consider a compact Lie group $G$ which acts on a closed Riemannian manifold $M$ by isometries. Then it is well-known that there are only finitely many isotropy types of the $G$-action, i.e. finitely ...

**7**

votes

**0**answers

115 views

### What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be:
Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...

**1**

vote

**1**answer

139 views

### Homomorphisms from irreducible spaces to reducible spaces

Let $P_{\lambda}$ be a Young symmetriser associated to the following tableau $(a_1 a_2 a_3 b_3 ; b_1 b_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take ...

**3**

votes

**0**answers

103 views

### Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...

**7**

votes

**2**answers

116 views

### Can monomial representations induced from nonmonomial representations?

Let $H$ be a subgroup of $G$. Let $\rho$ be an irr representation of $G$ induced from an irr representation $\theta$ of $H$. It is well known that $\rho$ is monomial if $\theta$ is monomial. Is it ...

**1**

vote

**0**answers

107 views

### Non invertibility of certain integral arising from group action

Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. ...

**2**

votes

**1**answer

76 views

### No basis change in a fusion ring allowed?

Consider the Fibonacci category F: $A\bigotimes{A}=E\bigoplus{A}$.
A study by Study (SCNR :-) already 1890 listed all unital associative algebras (with rank<=4).
...

**5**

votes

**2**answers

169 views

### Generators of invariant polynomials of semisimple Lie algebra

Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ ...

**2**

votes

**1**answer

90 views

### A quadratic algebra with four generators and four relations

Algebra that I'm going to describe pop-up in my research, it looks completely elementary, but I don't know any appropriate references.
Let $k$ be an algebraically closed field of characteristic ...

**2**

votes

**0**answers

64 views

### A question on Plancherel measure for $p$-adic group

Let $G$ be a reductive group over a $p$-adic field. My understanding is that the Plancherel measure on $G$ is a measure on the unitary dual $\hat{G}$.
But at the same time, for example, in his famous ...

**0**

votes

**0**answers

76 views

### G-equivariant coherent sheaves on Bott-Samelson Resolutions

Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...

**4**

votes

**2**answers

133 views

### Weight multiplicity formulae for $(\mathfrak g,B)$-irreps

Let $G$ be a complex reductive Lie group, $B$ a Borel subgroup, with which to define "dominant weight". Let $\lambda$ be an integral weight, not necessarily dominant, but nonetheless giving a ...

**1**

vote

**1**answer

153 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 ...

**2**

votes

**1**answer

97 views

### Number of double cosets of a Young subgroup

Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$.
Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...

**3**

votes

**1**answer

142 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 $\text{St}_{2n}(\mathbb{F}_q)$ ...

**2**

votes

**1**answer

121 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 ...

**10**

votes

**0**answers

182 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

119 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

109 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

112 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

47 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

353 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

67 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

**1**answer

127 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

74 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

116 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

214 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

124 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 ...

**10**

votes

**3**answers

617 views

### A class of matrix determinants between Wronskians and Vandermondes

Update: see below
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 ...