arXiv subjects

Gerald W. Schwarz

Publications and source records attributed to Gerald W. Schwarz.

31 records · Page 2Linked to original sources

arXiv2012

Vector Fields and Luna Strata

Let V be a G-module where G is a complex reductive group. Let Z:=V//G denote the categorical quotient. One can ask if the Luna stratification of Z is intrinsic. That is, if phi : Z\to Z is any automorphism, does phi send strata to strata? In a paper of Kuttler and Reichstein the answer was shown to be yes for V a direct sum of sufficiently many copies of a G-module W. We show that the answer is yes for almost all V. The key is to consider the vector fields on Z. Our methods also show that complex analytic automorphisms preserve the stratification.

math.GR↗

arXiv2012

Stratifications with respect to actions of real reductive groups

We study the action of a real reductive group G on a real submanifold X of a K"ahler manifold Z. We suppose that the action of G extends holomorphically to an action of a complex reductive group and is Hamiltonian with respect to a compatible maximal compact subgroup of the complex reductive group. There is a corresponding gradient map obtained from a Cartan decomposition of G. We obtain a Morse like function on X. Associated to its critical points are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds of X which are called pre-strata. In case that the gradient map is proper, the pre-strata form a decomposition of X and in case that X is compact they are the strata of a Morse type stratification of X. Our results are generalizations of results of Kirwan obtained in the case that X=Z is compact and the group itself is complex reductive.

math.CV↗

arXiv2011

Representations With a Reduced Null Cone

Let G be a complex reductive group and V a G-module. Let π: V \to V//G be the quotient morphism and set N(V) = π^{-1}(π(0)). We consider the following question. Is the null cone N(V) reduced, i.e., is the ideal of N(V) generated by G-invariant polynomials? We have complete results when G is SL_2, SL_3 or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible.

math.AG↗

arXiv2011

Reduced invariant sets

Let K be a compact Lie group and W a finite-dimensional real K-module. Let X be a K-stable real algebraic subset of W. Let I(X) denote the ideal of X in R[W] and let I_K(X) be the ideal generated by I(X)^K. We find necessary conditions and sufficient conditions for I(X)= I_K(X) and for \sqrt{I_K(X)}=I(X). We consider analogous questions for actions of complex reductive groups.

math.RT↗

arXiv2010

Linear maps preserving orbits

Let H\subset\GL(V) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let v\in V and let G=\{g\in\GL(V)\mid gHv = Hv\}. Following Raïs we say that the orbit Hv is \emph{characteristic for H} if the identity component of G is H. If H is semisimple, we say that Hv is \emph{semi-characteristic} for H if the identity component of G is an extension of H by a torus. We classify the H-orbits which are not (semi)-characteristic in many cases.

math.RT↗

arXiv2009

Isomorphisms preserving invariants

Let $V$ and $W$ be finite dimensional real vector spaces and let $G\subset\GL(V)$ and $H\subset\GL(W)$ be finite subgroups. Assume for simplicity that the actions contain no reflections. Let $Y$ and $Z$ denote the real algebraic varieties corresponding to $\R[V]^G$ and $\R[W]^H$, respectively. If $V$ and $W$ are quasi-isomorphic, i.e., if there is a linear isomorphism $L\colon V\to W$ such that $L$ sends $G$-orbits to $H$-orbits and $L\inv$ sends $H$-orbits to $G$-orbits, then $L$ induces an isomorphism of $Y$ and $Z$. Conversely, suppose that $f\colon Y\to Z$ is a germ of a diffeomorphism sending the origin of $Y$ to the origin of $Z$. Then we show that $V$ and $W$ are quasi-isomorphic, This result is closely related to a theorem of Strub \cite{Strub}, for which we give a new proof. We also give a new proof of a result of \cite{KrieglLosikMichor03} on lifting of biholomorphisms of quotient spaces.

math.RT↗

arXiv2009

Real double coset spaces and their invariants

Let G be a real form of a complex reductive group. Suppose that we are given involutions σand θof G. Let H=G^σdenote the fixed group of σand let K=G^θdenote the fixed group of θ. We are interested in calculating the double coset space H\backslash G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles.

math.RT↗

arXiv2008

Compression of Finite Group Actions and Covariant Dimension, II

Let $G$ be a finite group and $ϕ: V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $ϕ$ is faithful if $G$ acts faithfully on $ϕ(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\bar{ϕ(V)}$ taken over all faithful $ϕ$. In \cite{KS07} we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of \cite{KS07}, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps of \cite{KS07}.

math.AG↗

arXiv2008

Homology of equivariant vector fields

Let K be a compact Lie group. We compute the abelianization of the Lie algebra of equivariant vector fields on a smooth K-manifold X. We also compute the abelianization of the Lie algebra of strata preserving smooth vector fields on the quotient X/K.

math.DG↗

arXiv2008

Linear maps preserving fibers

Let $G\subset\GL(V)$ be a complex reductive group where $\dim V<\infty$, and let $π\colon V\to\quot VG$ be the categorical quotient. Let $\NN:=π\invπ(0)$ be the null cone of $V$, let $H_0$ be the subgroup of $\GL(V)$ which preserves the ideal $\I$ of $\NN$ and let $H$ be a Levi subgroup of $H_0$ containing $G$. We determine the identity component of $H$. In many cases we show that $H=H_0$. For adjoint representations we have $H=H_0$ and we determine $H$ completely. We also investigate the subgroup $G_F$ of $\GL(V)$ preserving a fiber $F$ of $π$ when $V$ is an irreducible cofree $G$-module.

math.GR↗

arXiv2007

Linear maps preserving invariants

Let $G\subset\GL(V)$ be a complex reductive group. Let $G'$ denote $\{ϕ\in\GL(V)\mid p\circϕ=p\text{for all} p\in\C[V]^G\}$. We show that, in general, $G'=G$. In case $G$ is the adjoint group of a simple Lie algebra $\lieg$, we show that $G'$ is an order 2 extension of $G$. We also calculate $G'$ for all representations of $\SL_2$.

math.RT↗

arXiv2006

When Polarizations Generate

Let G be a reductive complex algebraic group and V a finite-dimensional G-module. From elements of the invariant algebra C[V]^G we obtain by polarization elements of C[kV]^G, where k\geq 1 and kV denotes the direct sum of k copies of V. For G simple our main result is the classification of the G-modules V and integers k\geq 2 such that polarizations generate C[kV]^G.

math.RT↗

arXiv2006

Compression of Finite Group Actions and Covariant Dimension

Let $G$ be a finite group and $ϕ\colon V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $ϕ$ is faithful if $G$ acts faithfully on $ϕ(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\bar{ϕ(V)}$ taken over all faithful $ϕ$. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify groups of covariant dimension less than or equal to 2. A byproduct of our investigations is the existence of a purely transcendental field of definition of degree $n-3$ for a generic field extension of degree $n\geq 5$.

math.RT↗