Transformation Groups最新文献

We present an elementary proof of the fact that every torsor under an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of fibre functors on neutral Tannakian categories.

We prove the stability conjecture of (imath )canonical bases, which was raised by Huanchen Bao and Weiqiang Wang in 2016, for all locally finite types. To this end, we characterize the trivial module over the (imath )quantum groups of such type at (q = infty ). This result can be seen as a very restrictive version of the (imath )crystal base theory for locally finite types.

Let G be a split connected reductive group over (mathbb {F}_q) and let (mathbb {P}^1) be the projective line over (mathbb {F}_q). Firstly, we give an explicit formula for the number of (mathbb {F}_{q})-rational points of generalized Steinberg varieties of G. Secondly, for each principal G-bundle over (mathbb {P}^1), we give an explicit formula counting the number of triples consisting of parabolic structures at 0 and (infty ) and a compatible nilpotent section of the associated adjoint bundle. In the case of (GL_{n}) we calculate a generating function of such volumes re-deriving a result of Mellit.

Let G be a connected reductive group split over (mathbb R). We show that every unipotent element in the totally nonnegative monoid of G is regular in some Levi subgroups, confirming a conjecture of Lusztig.

We prove that the zero-fiber of the moment map of a totally negative quiver has rational singularities. Our proof consists in generalizing dimension bounds on jet spaces of this fiber, which were introduced by Budur. We also transfer the rational singularities property to other moduli spaces of objects in 2-Calabi-Yau categories, based on recent work of Davison. This has interesting arithmetic applications on quiver moment maps and moduli spaces of objects in 2-Calabi-Yau categories. First, we generalize results of Wyss on the asymptotic behaviour of counts of jets of quiver moment maps over finite fields. Moreover, we interpret the limit of counts of jets on a given moduli space as its p-adic volume under a canonical measure analogous to the measure built by Carocci, Orecchia and Wyss on certain moduli spaces of coherent sheaves.

We prove that every filtered fiber functor on the category of dualizable representations of a smooth affine group scheme with enough dualizable representations comes from a graded fiber functor.

We determine the existence of cocompact lattices in groups of the form (textrm{V}rtimes textrm{SL}_2(mathbb {R})), where (textrm{V}) is a finite dimensional real representation of (textrm{SL}_2(mathbb {R})). It turns out that the answer depends on the parity of (dim (textrm{V})) when the representation is irreducible.

Let (textrm{E}=textrm{E}_{bar{0}}oplus textrm{E}_{bar{1}}) be a real or complex (mathbb {Z}_2)-graded vector space equipped with an even supersymmetric bilinear form that restricts to a symplectic form on (textrm{E}_{bar{0}}) and an orthogonal form on (textrm{E}_{bar{1}}). We obtain a full classification of reductive dual pairs in the (real or complex) orthosymplectic Lie superalgebra (mathfrak {spo})(E) and its associated Lie supergroup ({textbf {SpO}}(textrm{E})). Similar to the purely even case, dual pairs are divided into two subclasses: Type I and Type II. The main difference with the purely even case occurs in the characterization of (super)hermitian forms on modules over division superalgebras. We then use this classification to prove that for a reductive dual pair ((mathscr {G},, mathscr {G}') = ((textrm{G},, mathfrak {g}),, (textrm{G}',, mathfrak {g}'))) in ({textbf {SpO}}(textrm{E})), the superalgebra ({textbf {WC}}(textrm{E})^{mathscr {G}}) that consists of (mathscr {G})-invariant elements in the Weyl-Clifford algebra ({textbf {WC}}(textrm{E})), when it is equipped with the natural action of the orthosymplectic Lie supergroup ({textbf {SpO}}(textrm{E})), is generated by the Lie superalgebra (mathfrak {g}'). As an application of the latter double commutant property, we prove that Howe duality holds for the dual pairs (( {{textbf {SpO}}}(2n|1),, {{textbf {OSp}}}(2k|2l)) subseteq {{textbf {SpO}}}(mathbb {C}^{2k|2l} otimes mathbb {C}^{2n|1})).

In this article, we consider certain irreducible subvarieties of the moduli space of compact Riemann surfaces determined by the specification of actions of finite groups. We address the general problem of determining which among them are non-normal subvarieties of the moduli space. We obtain several new examples of subvarieties with this property.

We use combinatorics of qq-characters to study extensions of deformed W-algebras. We describe additional currents and part of the relations in the cases of (mathfrak {gl}(n|m)) and (mathfrak {osp}(2|2n)).