**8**

votes

**2**answers

229 views

### Integrals of representations over geodesics

Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the ...

**1**

vote

**1**answer

221 views

### Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y ...

**8**

votes

**3**answers

180 views

### How do small central extensions drop the dimension of a faithful representation?

Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied...
I am interested in the phenomenon ...

**6**

votes

**5**answers

546 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 ...

**7**

votes

**2**answers

445 views

### Good effective versions of theorems of Artin and Brauer

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups.
For example, Artin's theorem is the statement that for every character $\chi$ of ...

**5**

votes

**0**answers

81 views

### Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B

This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...

**0**

votes

**0**answers

93 views

### Anticommuting operators with positive properties

Which classes of $M\in \mathsf M_k(\Bbb R)^{n\times n}$ admit solutions $N\in \mathsf M_k(\Bbb R)^{n\times n}$ such that
$$(M\otimes N+N\otimes M)(u\otimes u)=0$$ forall $u\in \mathsf D_k(\Bbb ...

**2**

votes

**1**answer

280 views

### Canonical representation of $\operatorname{SL}(2,\mathbb{R})$ on $L^2(\mathbb{R}^2)$

As a unimodular subgroup of the group of automorphisms of $\mathbb{R}^2$, $\operatorname{SL}(2,\mathbb{R})$ can be represented as a subgroup of $\mathcal{U}(L^2(\mathbb{R}^2))$ (the group of unitary ...

**2**

votes

**0**answers

73 views

### Adding a row to a Young Tableau via Novelli-Pak-Stoyanovskii

Let $T_{\lambda}$ be the set of standard young tableaux (SYT) of shape $\lambda_1\geq \lambda_2\cdots\geq \lambda_n$. Now consider pushing a row $\mu$ with $\mu\geq \lambda_1$ onto $Y$ to give shape ...

**1**

vote

**0**answers

156 views

### What is the spectrum of $L^1(G:H)$?

Let $H$ be a compact subgroup of a locally compact topological group $G$ and
$$ L^1(G:H)=\{f\in L^1(G): R_h f=f\;\text{ a.e. }\; \forall h \in H\}$$ and $\widehat{(G:H)}=\{\xi\in ...

**2**

votes

**3**answers

513 views

### Computing the Grothendieck-Springer resolution for $G = SL_2$

Let $G = SL_2, \mathfrak{g} = \mathfrak{sl}_2$, $B$ the Borel subgroup, and $\mathfrak{u}$ the unipotent radical; so that $G/B = \mathbb{P}^1$; how does $\widetilde{\mathfrak{g}}$ decompose as a ...

**5**

votes

**0**answers

67 views

### How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

**0**

votes

**1**answer

147 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} $, ...

**5**

votes

**0**answers

84 views

### Wavefront sets of irreducible representations with non-integral infinitesimal characters

Let $G$ be a complex reductive algebraic group (connected, simply connected etc), viewed as a real group. We study the representations of $G$, and we follow the notations in the paper of Barbasch and ...

**28**

votes

**5**answers

5k views

### Definitions of Hecke algebras

There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are ...

**7**

votes

**1**answer

236 views

### Division algebras over extension fields / reducibility of $G$-modules

Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...

**0**

votes

**0**answers

59 views

### Maximum symmetry of generic 6j symbols?

I can't get those three papers collinear...
The "classic" general reference is Butler (including conjugates and multiplicities). An example from a spherical category is Hong, an example from a ...

**2**

votes

**0**answers

246 views

### A dual version of a theorem of Øystein Ore in group theory

This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: Let $(H \subset G)$ be an inclusion of finite groups such that the lattice ...

**4**

votes

**2**answers

275 views

### Concise mathematical definition of the fusion product on the Verlinde ring?

The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the ...

**8**

votes

**0**answers

473 views

### Strange boundary-like map on tensor algebra: what is its kernel?

Let $k$ be a commutative ring and $L$ a $k$-module. The tensor algebra $\otimes L$ is $\mathbb{Z}$-graded and $\mathbb{Z}_2$-graded (an element of $L^{\otimes n}$ has degree $n$ and ...

**6**

votes

**1**answer

395 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 ...

**8**

votes

**1**answer

310 views

### Automorphisms of generic complete intersections

This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions.
To state the question, let $r \geq ...

**2**

votes

**0**answers

85 views

### What's the relation of the Hecke algebra of a pair and the flag variety?

Let $G$ be a real semisimple Lie group and $K$ a maximal compact subgroup. Let $\mathfrak{g}$ and $\mathfrak{k}$ be the complexified Lie algebra of $G$ and $K$, respectively.
Then the Hecke algebra ...

**1**

vote

**1**answer

68 views

### Explicit deformations of pseudo representations

Let $G$ be a group (which I will be glad to consider to be the absolute Galois group of a $p$-adic field, and so satisfies Mazur's finiteness condition which appears in his paper Deforming Galois ...

**2**

votes

**1**answer

111 views

### cohomology of orthogonal (or general linear) group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let
$$
O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\}
$$
What is $$
...

**2**

votes

**0**answers

99 views

### Free-field representations: how to study highest-weight submodules of the Fock module?

Suppose we have a representation of some affine Lie algebra $\mathfrak{g}=\mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$ on a Fock space $V$. The module $V$ will contain a lot of ...

**17**

votes

**3**answers

552 views

### Center of a simply-connected simple compact Lie group and McKay correspondence

Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$.
I find the following interpretation of $Z(G)$ in ...

**10**

votes

**1**answer

329 views

### Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

**6**

votes

**1**answer

189 views

### In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective?
Recall that an object ...

**4**

votes

**0**answers

123 views

### Counting points on Hessenberg varieties over a finite field

Let $G$ be an connected reductive group over finite field $k$. I will assume that $\text{char}(k)$ is very good for $G$ (or even larger, if preferred). Let $B\subset G$ be a Borel subgroup defined ...

**6**

votes

**2**answers

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

**3**

votes

**2**answers

300 views

### symmetric 2-cocycle / many projective representations

Let $G$ be a finite group, $k$ the field of complex numbers.
Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$:
...

**2**

votes

**2**answers

101 views

### admissible characters for $PGL_{2}(F)$

What are the irreducible admissible representations of $PGL_{2}(F)$ for $F$ a local nonarchimedean field and do we have formulas for their characters?

**2**

votes

**0**answers

50 views

### explicit matrices for Weil ($p^2$ dimensional) representation of $Sp(4,\mathbb{F}_p)$, $p>3$

I am looking for more-or-less explicit matrices for the $p^2$ dimensional Weil representation of $Sp(4,\mathbb{F}_p)$, suitable for computer implementation. Ideally, I would like the images of the ...

**10**

votes

**1**answer

658 views

### Number of standard Young tableaux with fixed corner entry

For a partition $\lambda=(\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_k)$ of $n$, let the set of standard Young tableau of shape $\lambda$ be denoted by $SYT(\lambda)$ with boxes at $(i,j)$ denoted ...

**13**

votes

**2**answers

648 views

### What's the status of Arthur's announced classification for GSp(4)?

In "Automorphic representations of GSp(4)" (2004) (see http://www.math.toronto.edu/arthur/), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no ...

**1**

vote

**1**answer

166 views

### cohomology of orthogonal group of integers

Let
$$
O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k).
$$
What is $$
H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})?
$$
If it cannot be computed out, can we get
$$
H^*(O(\mathbb{Z}^{\oplus ...

**3**

votes

**0**answers

108 views

### The Tangent Bundle of the Space of CR Structures on S^(2n+1)

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...

**3**

votes

**0**answers

61 views

### Linear independence of points under projection of Veronese re-embedding

Let $V$ be a complex vector space.
Let $x_1,...,x_k\in PV$. Let $v_d: PV\rightarrow PS^dV$
be the Veronese. Then $v_d(x_1),...,v_d(x_k)$ are in general linear position
as long as $k\leq d-1$.
Now let ...

**8**

votes

**1**answer

1k 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
...

**0**

votes

**0**answers

62 views

### On continuous part of the L^2 spectrum

Suppose $G$ is a real reductive Lie group and $\Gamma$ is a lattice in $G$ (of finite co-volume). I am reading Langlands's paper " On the functional equation satisfied by the Eisenstein series". I ...

**1**

vote

**0**answers

110 views

### counting how many boxes from a given Young tableau contribute to hook length made out of two YTs

Think of a Young diagram as a collection of rows with numbers of elements
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...

**2**

votes

**1**answer

155 views

### Relation between Different Definitions of Induced Representation

I've seen two different ways to define induced representation.
One is as in the book Introduction to representation theory: If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of ...

**3**

votes

**1**answer

249 views

### Number of semi-standard tableau

What is the number of semi-standard tableau (weakly increasing on rows and strictly increasing on columns) for the partition $2n=n+n$ with entries $\{1,2, \cdots ,n\}$ such that each $i$ appears ...

**0**

votes

**0**answers

34 views

### Ten dimensional representation of $S_6$ [migrated]

Let $S=\{1,2,3,4,5,6\}$.
For every three-element subset $A\subset S$ and $B=S\setminus A$ consider the symbol $e_{(A|B)}$ for which we assume that $e_{(A|B)}=e_{(B|A)}$. Then the vector space $V$ ...

**0**

votes

**0**answers

75 views

### Central extensions of SL2(R) by U(1) ?

Can somebody please tell me what are the central extensions of SL2(R) by U(1), that is, what is $H^2(SL2(R), U(1)) $ ? Thank you

**2**

votes

**0**answers

67 views

### Dimension of affine Springer fiber and its functor of points as an ind-scheme

Let $k$ be a finite field and let $F = k( (t))$ with ring of integers $\mathfrak{o} = k[ [t]]$. Let $G$ be a connected linear algebraic $k$-group with Lie algebra $\mathfrak{g}$. Suppose that ...

**19**

votes

**1**answer

424 views

### How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity
To ...

**6**

votes

**1**answer

215 views

### Jordan decomposition of the tensor product of two matrices

I asked this question on Math.SE here, but did not get a lot of attention.
I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...

**6**

votes

**0**answers

213 views

### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...