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Register a new userHolomorphic harmonic analysis on complex reductive groups
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We define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties like the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of $K$-admissible measures. We prove that $K$-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of $K$-admissible measures.
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We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k be a purely inseparable field extension of k of degree p^e and let G denote the Weil restriction of scalars R_{k/k}(G) of a reductive k-group G. When G= R_{k/k}(G) we also provide some results on the orders of elements of the unipotent radical RR_u(G_{bar k}) of the extension of scalars of G to the algebraic closure bar k of k.
Over a $p$-adic local field $F$ of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group $G={mathbb G}_mtimes{mathrm Sp}_{2n}$. It is associated to the Langlands $gamma$-functions attached to any irreducible admissible representations $chiotimespi$ of $G(F)$ and the standard representation $rho$ of the dual group $G^vee({mathbb C})$, and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on ${rm GL}_1(F)$, which is associated to a $gamma$-function $beta_psi(chi_s)$ (a product of $n+1$ certain abelian $gamma$-functions). Our work on ${rm GL}_1(F)$ plays an indispensable role in the development of our work on $G(F)$. These two types of harmonic analyses both specialize to the well-known local theory developed in Tates thesis when $n=0$. The approach is to use the compactification of ${rm Sp}_{2n}$ in the Grassmannian variety of ${rm Sp}_{4n}$, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis and many other works) on the doubling local zeta integrals for the standard $L$-functions of ${rm Sp}_{2n}$. The method can be viewed as an extension of the work of Godement-Jacquet for the standard $L$-function of ${rm GL}_n$ and is expected to work for all classical groups. We will consider the archimedean local theory and the global theory in our future work.
Let $G$ be a simple algebraic group of type $G_2$ over an algebraically closed field of characteristic $2$. We give an example of a finite group $Gamma$ with Sylow $2$-subgroup $Gamma_2$ and an infinite family of pairwise non-conjugate homomorphisms $rhocolon Gammarightarrow G$ whose restrictions to $Gamma_2$ are all conjugate. This answers a question of Burkhard Kulshammer from 1995. We also give an action of $Gamma$ on a connected unipotent group $V$ such that the map of 1-cohomologies ${rm H}^1(Gamma,V)rightarrow {rm H}^1(Gamma_p,V)$ induced by restriction of 1-cocycles has an infinite fibre.
Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an appropriate sense. We generalise this idea to arbitrary reductive $G$ using the notion of $G$-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a $G$-completely reducible subgroup $H$ of $G$, unique up to $G(k)$-conjugacy, which we call a $k$-semisimplification of $H$. This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL_n$ and with Serres $G$-analogue of semisimplification for subgroups of $G(k)$). We also show that under some extra hypotheses, one can pick $H$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf and Rousseau.
Let $G$ be a reductive algebraic group over an algebraically closed field and let $V$ be a quasi-projective $G$-variety. We prove that the set of points $vin V$ such that ${rm dim}(G_v)$ is minimal and $G_v$ is reductive is open. We also prove some results on the existence of principal stabilisers in an appropriate sense.
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