All Questions
Tagged with rt.representation-theory lie-groups
832 questions
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Collection of matrices in $SU_{\mathbb{C}}(n)$ with given family of eigenvectors
For a given fixed matrix $M\in SU_{\mathbb{C}}(n)$, how to find all $N\in SO_{\mathbb{C}}(n)$ such that $N^{-1}MN$ is a diagonal matrix?
If we consider a fixed set of $n$ complex vectors $\Gamma:=\{...
- 8,071
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1
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232
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Large spin expansion of affine $\mathfrak{su}(2)_k$ characters
There is a problem I am trying to solve for some time now which in a few words boils down to computing the coset characters for
$$
\frac{\mathfrak{su}(2)_k\oplus\mathfrak{su}(2)_\ell}{\mathfrak{su}(2)...
- 71
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264
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labeling state vectors in representation space of a simple lie algebra
Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number
of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation space....
- 85
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608
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Para-Complexification of Lie Groups
Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...
user21574
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Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)
I have difficulties finding an appropriate reference for the following question (which I hope that it to be true).
Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexification and $\tau: U^...
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1
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222
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Degree bounds when restricting an irrep of a compact Lie group to a torus
I am not sure of the right terminology, but here goes. Let $G$ be a compact, connected, simply connected, non-abelian Lie group.
For any choice of one-dimensional torus $S\subset G$, and any finite-...
- 25.8k
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173
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Representing elements of U(N) or SO(N) by elementary rotations exp(i phi_n sigma_n)
Given a (orthogonal) basis $(\sigma_n)_{n=1,\dots,K}$ of the algebra $u(N)$, we can represent any element $U$ of the corresponding group $U(N)$ in the form
$U=e^{i\sum_{n=1}^K\varphi_n\sigma_n}$.
Is ...
- 13
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489
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generalisation of GL(3,R) polar decomposition
Does polar decomposition work when the 'orthogonal' matrices are not orthogonal wrt to the identity ($O^TO=Id$), but wrt to some other symmetric matrix $K$ (i.e. $O^TKO=K$)?
Specifically, $GL(3,R)$ ...
- 11
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71
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Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
- 6,622
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72
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Normalizer of connected subgroup contained in the Weyl group?
Let $ G $ be a simple Lie group. Let $ H $ be a connected subgroup of $ G $ such that $ N(H)/H $ is finite. In such a case, is $ N(H)/H $ always a subgroup of the Weyl group of $ G $?
For $ G=\...
- 4,081
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108
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Schur Weyl Duality for Maximal Torus
I wanted to know if there's some version of Schur Weyl Duality for the maximal torus $T \subset \operatorname{GL}(V)$? Is there also some combinatorial object which might be useful for the same?
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What is the form of the incomplete Eisenstein series on PGL_2(C)?
Let $F$ be an imaginary quadratic number field. Let $G = \mathrm{PGL}_2(\mathbb{C}) $ and $\varGamma = \mathrm{PSL}_2(\mathcal{O}_F)$. We have the Iwasawa decomposition $G = NAK$ where $K = \mathrm{...
- 185
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113
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A combinatoric identity for characters of reductive groups
Let $G$ be a reductive group over an algebraic closed field (of char 0 if necessary). Let $T\subset G$ be a maximal torus and $S=\mathrm{Sym}^*(X(T))$ be the symmetric algebra of characters of $T$. ...
- 11
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70
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Minimal $K$-orbit on $\mathfrak{g}$
Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra with Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. Write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the ...
- 951
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Weight of adjoint action on a lower central series extension
Let $\mathcal{U}$ be a unipotent Lie $\mathbb{Q}_p$-group scheme, whose associated gradeds from the lower central series filtration are $\mathcal{U}_0 = \mathcal{U}^{\text{ab}}$, $\mathcal{U}_1 = [\...
- 2,907
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65
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Classical groups generated by tensor products of subgroups
Let $ G $ denote a classical group.
Question:
Is it the case that
$$
\langle G_n \otimes G_m,G_m \otimes G_n\rangle=G_{nm}
$$
as long as $ n \neq m $?
For example, if $ G $ is the classical group $ GL(...
- 4,081
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119
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Question on Artin's Gamma function on $\operatorname{SO}(2,0)(\mathbb R)$
$\DeclareMathOperator\SO{SO}$Let $G=\SO(2,0)(\mathbb{R})$, a quasi-split group with signature $(2,0)$. Let $e$ be an element in $O(2,0)(\mathbb{R}) \setminus \SO(2,0)(\mathbb{R})$.
Let $\pi$ be an ...
- 1,019
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105
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Weyl group action on the Lie algebra [duplicate]
Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...
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70
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Orbit projection geometry
Background:
As shown in [1] and [2], for a closed smooth submanifold $M$ of $\mathbb R^d$, the domain $D_M$ of the projection map $P_M:D_M\rightarrow M$ has a dense interior $\Omega_M$ over which $P_M|...
- 71
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K-finiteness of unitary representations of Poincaré-like groups?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\ISO{ISO}$I'd like to know if there are any papers that study the following problems:
Determine when decomposing the unitary irreps of $\ISO(d,1)$ into ...
- 144
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201
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On the center of the universal enveloping algebra of a Lie algebra
Consider a finite-dimensional Lie algebra over C. Sometimes (e.g. for any semisimple Lie algebra) the center of its universal enveloping algebra is isomorphic to a freely generated polynomial ring.
...
- 11
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244
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Relation between projective representation and the representation of the universal cover of a Lie Group
I am reading this paper, in what says exactly:
"Weare dealing with a ray representation os the conformal group AND THEREFORE with a representation of the universal covering group of the conformal ...
- 424
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138
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When is the zero weight space of an irreducible $\frak{sl}_{n+1}$-module non-trivial?
Take the complex semisimple Lie algebra $\frak{sl}_{n+1}$, with space of dominant integral weights $P(\frak{sl}_{n+1})$. For $V(\lambda)$ the irreducible representation corresponding to $\lambda \in P(...
- 969
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133
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Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
- 1,121
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125
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A basic application of Mackey's theorem
Let $G=GL(2,\mathbb {F}_q)$ and $B=\left\{\begin{pmatrix}
* & * \\
0 & *
\end{pmatrix}\right\}$ the
Borel subgroup of upper triangular matrices. Let $\chi_1$ and $\chi_2$ be the characters ...
- 155
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102
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Bounding the dimensions of faithful representations of a quotient group
For $G$ a compact Lie group, let $\operatorname{mdfr}(G)$ be the minimum dimension of a faithful complex representation of $G$. Is there a bound on $\operatorname{mdfr}(N(H)/H)$ for $H$ a subgroup of ...
- 201
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92
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The $U({\frak g})v$-module generated by a single element of a $U({\frak g})v$-module
Let $\frak{g}$ be a finite dimensional complex semisimple Lie algebra and let $U(\frak{g})$ be its universal enveloping algebra. Take $V$ an infinite dimensional module over $U(\frak{g})$. Let $v \in ...
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285
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Coadjoint orbits
I've posted the following question some days ago in math.stackexchange https://math.stackexchange.com/questions/4155747/co-adjoint-orbit but I didn't get any answer!
While I was trying to teach my ...
- 139
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What subspace of $\operatorname{SU}(4)$ group keeps an element of the $\mathfrak{su}(2)$ subalgebra within $\mathfrak{su}(2)$ upon adjoint action?
Consider the Lie group $G_4=\operatorname{SU}(4)$ with (15) generators $T^a$. A basis for the latter is
$$\{\sigma^j \times 1_2, \quad \quad \sigma^i \times \sigma^j, \quad \quad 1_2 \times \sigma^j\},...
- 155
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133
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What is the analogue of Leibniz's rule for universal enveloping algebra?
Let $G$ be a reductive group over $\mathbb{R}$ and $\mathfrak{g}$ its complexitied Lie algebra.
Let $U(\mathfrak{g})$ be the universal enveloping algebra and $Z(\mathfrak{g})$ is the center of $U(\...
- 1,759
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Why is this operator independent of the choice of basis
I asked this question in MSE but I received no answer
https://math.stackexchange.com/questions/4009524/why-is-the-following-operator-independent-of-the-choice-of-basis/4013636#4013636
Let $G$ be a lie ...
- 139
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142
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Principal orbit and the generic stabilizer of SO(2n)xSO(2n)
Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers.
Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, ...
- 11
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62
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Circle representations and regular representations
Write $\mathbb T$ for the circle group, $C_n$ for the cyclic group of order $n$, and fix embeddings $C_n \hookrightarrow \mathbb T$. Let $\lambda^i$ be the $\mathbb T$-representation with underlying ...
- 1,838
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100
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Embedding of discrete series
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals that of its maximal compact subgroup. Let $G'$ be a reductive subgroup of $G$ with equal rank. If $\tau$ is a discrete ...
- 951
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103
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Which operators constructed from 10d gamma matrices commute with $SO(1,2)\times SO(3)\times SO(3)$?
In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in ...
- 1,155
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100
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Representation equivalent lattices
Suppose $G$ is a absolutely almost simple algebraic groups over a number field $K$. Let $\Gamma_1$ and $\Gamma_2$ be two lattices in $G(K)$. Then $\Gamma_1$ and $\Gamma_2$ are said to be ...
- 61
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84
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Integrable modules and comodules
Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...
- 6,201
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159
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Notation clash between a representation and spectral radius
I am currently writing a paper where I need talk both about a representation of a semisimple Lie group (usually denoted by $\rho$), and about spectral radii of linear maps (also usually denoted by $\...
- 1,574
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70
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$\Gamma$ cohomology of principal series
Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup
with Langlands ...
- 1,111
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71
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Computing equivariant K-theory using the amalgamted product
If I have a Lie group (or a Kac-Moody group) $G$ such that it's the amalgated product of it's proper parabolic subgroups $P_J$, i.e. $G = \text{colim} P_J$, then could I use this to compute the ...
- 257
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110
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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 ...
- 61
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197
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exponential and anisotropic torus
Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...
- 3,472
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196
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Reference Help: Matsuki duality Orbits
I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...
- 111
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85
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Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group
Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...
- 1,582
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2
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360
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When representation of two different coadjoint orbits are equivalent?
Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where $\mathfrak{t}$...
user21574
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0
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508
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on the open bruhat cell
Let $G$ a connected reductive group and $S=U^{-}TU$ the open cell.
Do we have $G=\bigcup\limits_{g\in G}gSg^{-1}$?
And also if I assume that $G$ is adjoint and $\overline{G}$ is the de Concini-...
- 3,472
1
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2
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469
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Finding spherical representations of $GL(n, \mathbb{C})$.
I am looking for literature that might contain the spherical representations of $GL(n, \mathbb{C})$. Here a spherical representation is an irreducible representation $\rho$ of $G$ on $\mathbb{C}$ such ...
- 247
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0
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158
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Zariski dense subgroup of $SL(3,\mathbb{R})$
Let $\Delta$ be a Zariski dense finitely generated subgroup of $SL(3,\mathbb{R})$. Assume that $\Delta$ contains no element of finite order. Then, does there exist a finite-order element $A \in SL(3,\...
- 133
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218
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Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.
I have a few questions on an application of the Weyl character formula.
To start with we work with the $\mathbb{Q}$ version of Hamilton's quaternions and consider the maximal order $\mathfrak{O} = \...
- 562
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66
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topologies on globalizations
I am reading notes by David Vogan on Unitary representations and Complex analysis (pdf / dvi).
The setting is as follows (page 23): Let $X$ be a $(\mathfrak{g},K)$-module and let $X(\mu)$ denote its ...
- 8,597