Prehomogeneous vector spaces and field extensions III
英文摘要:
In this paper, we determine the rational orbit decomposition for two prehomogeneous vector spaces associated with the simple group of type G_2.
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Akihiko Yukie
In this paper, we determine the rational orbit decomposition for two prehomogeneous vector spaces associated with the simple group of type G_2.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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Let H_1=SL(5), H_2=SL(3), H=H_1 \times H_2. It is known that (G,V) is a prehomogeneous vector space (see [22], [26], [25], for the definition of prehomogeneous vector spaces). A non-constant polynomial δ(x) on V is called a relative invariant polynomial if there exists a character χsuch that δ(gx)=χ(g)δ(x). Such δ(x) exists for our case and is essentially unique. So we define V^{ss}={x in V such that δ(x) is not equal to 0}. For x in V_R^{ss}, let H_{x R+}^0 be the connected component of 1 in classical topology of the stabilizer H_{x R}. We will prove that if x in V_R^ss is "sufficiently irrat…
The purpose of the paper is to prove an analogous correspondence for the above prehomogeneous vector space (1)-(3). For case (1) the correspondence is bijective. However, it turns out that the correspondence is not bijective for cases (2), (3). We describe the fiber structures of this correspondence in subsection 2-4. In section 1 we briefly review basic properties of the non-abelian Galois cohomology set and its relation to the orbit space G_k \ V_k^ss. In subsection 2-4 we consider the prehomogeneous vector spaces (1)-(3) respectively.
Let $G_\C$ be a connected, linear algebraic group defined over $\R$, acting regularly on a finite dimensional vector space $V_\C$ over $\C$ with $\R$-structure $V_\R$. Assume that $V_\C$ posseses a Zariski-dense orbit, so that $(G_\C,ρ,V_\C)$ becomes a prehomogeneous vector space over $\R$. We consider the left regular representation $π$ of the group of $\R$-rational points $G_\R$ on the Banach space $\Cvan(V_\R)$ of continuous functions on $V_\R$ vanishing at infinity, and study the convolution operators $π(f)$, where $f$ is a rapidly decreasing function on the identity component of $G_\R$. D…
Let $k$ be a field of characteristic not equal to $2,3$, $\mathbb{O}$ an octonion over $k$ and $\mathcal{J}$ the exceptional Jordan algebra defined by $\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\times \mathrm{GL}(2)$ and $V=\mathcal{J}\oplus \mathcal{J}$. We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with $k$-isomorphism classes of pairs $(\mathcal{M},\mathfrak{n})$ where $\mathcal{M}$'s are isotopes of $\mathcal{J}$ and $\mathfrak{n}$'s are cubic étale subalgebras of $\mathcal{M}$. Also we prove th…
Let $G$ be a connected reductive algebraic group and $B$ be a Borel subgroup defined over an algebraically closed field of characteristic $p>0$. In this paper, the authors study the existence of generic $G$-cohomology and its stability with rational $G$-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of $G$ and $B$-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of $G({\mathbb F}_{q})$ are established which generalizes earlier work of Hiller, in addition to stability ranges for gener…
Following work by Casazza, Kutyniok and Lammers, and extensions by Stoeva and Christensen, we provide some novel characterizations of R-dual sequences of type III in Hilbert spaces. We systematically extend the construction procedure by basing it on a choice of an antiunitary involution. For certain classes of R-duals of type III, we derive a representation of the associated frame operator in terms of spectral measures.
Let $F$ be a nonarchimedean local field, let $E$ be a Galois quadratic extension of $F$ and let $G$ be a quasisplit group defined over $F$; a conjecture by Dipendra Prasad states that the Steinberg representation of $G(E)$ is then $χ$-distinguished for a given unique character $χ$ of $G(F)$. In the first two papers of the series, Broussous and the author have proved that result when $G$ is $F$-split and $E/F$ is unramified; this paper deals with the tamely ramified case, still with $G$ $F$-split.
This is the third paper in the series. Here we define a few combinatorial orders on Young tableaux. The first order is obtained from induced Duflo order by the extension with the help of Vogan T_{α, β} procedure. We call it Duflo-Vogan order. The second order is obtained from the generalization of Spaltenstein's construction by consideration of an orbital variety as a double chain of nilpotent orbits. We call it the chain order. Again, we use Vogan's T_{α, β} procedure, however, this time to restrict the chain order. We call it Vogan-chain order. The order on Young tableaux defined by the incl…
For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q = 2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Moebius group of conformal transformations on R^{p-1, q-1}, and preserves a space of solutions of the ultrahyperbolic Laplace equation on R^{p-1, q-1}. We construct in an intrinsic and natural way a Hilbert space of ultrahyperbolic solutions so that O(p,q) becomes a continuous irre…
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