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Register a new userGraded unipotent groups and Grosshans theory
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Let $U$ be a unipotent group which is graded in the sense that it has an extension $H$ by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of $U$ are strictly positive. We study embeddings of $H$ in a general linear group $G$ which possess Grosshans-like properties. More precisely, suppose $H$ acts on a projective variety $X$ and its action extends to an action of $G$ which is linear with respect to an ample line bundle on $X$. Then, provided that we are willing to twist the linearisation of the action of $H$ by a suitable (rational) character of $H$, we find that the $H$-invariants form a finitely generated algebra and hence define a projective variety $X/!/H$; moreover the natural morphism from the semistable locus in $X$ to $X/!/H$ is surjective, and semistable points in $X$ are identified in $X/!/H$ if and only if the closures of their $H$-orbits meet in the semistable locus. A similar result applies when we replace $X$ by its product with the projective line; this gives us a projective completion of a geometric quotient of a $U$-invariant open subset of $X$ by the action of the unipotent group $U$.
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Let $U$ be a graded unipotent group over the complex numbers, in the sense that it has an extension $hat{U}$ by the multiplicative group such that the action of the multiplicative group by conjugation on the Lie algebra of $U$ has all its weights strictly positive. Given any action of $U$ on a projective variety $X$ extending to an action of $hat{U}$ which is linear with respect to an ample line bundle on $X$, then provided that one is willing to replace the line bundle with a tensor power and to twist the linearisation of the action of $hat{U}$ by a suitable (rational) character, and provided an additional condition is satisfied which is the analogue of the condition in classical GIT that there should be no strictly semistable points for the action, we show that the $hat{U}$-invariants form a finitely generated graded algebra; moreover the natural morphism from the semistable subset of $X$ to the enveloping quotient is surjective and expresses the enveloping quotient as a geometric quotient of the semistable subset. Applying this result with $X$ replaced by its product with the projective line gives us a projective variety which is a geometric quotient by $hat{U}$ of an invariant open subset of the product of $X$ with the affine line and contains as an open subset a geometric quotient of a U-invariant open subset of $X$ by the action of $U$. Furthermore these open subsets of $X$ and its product with the affine line can be described using criteria similar to the Hilbert-Mumford criteria in classical GIT.
The aim of this paper is to show that classical geometric invariant theory (GIT) has an effective analogue for linear actions of a non-reductive algebraic group $H$ with graded unipotent radical on a projective scheme $X$. Here the linear action of $H$ is required to extend to a semi-direct product $hat{H} = H rtimes mathbb{G}_m$ with a multiplicative one-parameter group which acts on the Lie algebra of the unipotent radical $U$ of $H$ with all weights strictly positive, and which centralises a Levi subgroup $R cong H/U$ of $H$. We show that $X$ has an $H$-invariant open subscheme (the hat-stable locus) which has a geometric quotient by the $H$-action. This geometric quotient has a projective completion which is a categorical quotient (indeed, a good quotient) by $hat{H}$ of an open subscheme of a blow-up of the product of $X$ with the affine line; with additional blow-ups a projective completion which is itself a geometric quotient can be obtained. Furthermore the hat-stable locus of $X$ and the corresponding open subsets of the blow-ups of the product of $X$ with the affine line can be described effectively using Hilbert-Mumford-like criteria combined with the explicit blow-up constructions. Applications include the construction of moduli spaces of sheaves and Higgs bundles of fixed Harder--Narasimhan type over a fixed nonsingular projective scheme, and of moduli spaces of unstable projective curves of fixed singularity. More recently, cohomology theory for reductive GIT quotients were extended by the first and fourth author to the non-reductive situation studied in this paper, and this was used to prove the Green--Griffiths--Lang and Kobayashi hyperbolicity conjectures for generic projective hypersurfaces with polynomial bounds on their degree.
We define a general notion of centrally $Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $Gamma$ is an arbitrary (generalized) ring. The case $Gamma$ = Z/2Z is studied in detail: it is related to Clifford algebras and their discrete Clifford groups (also called Salingaros Vee groups).
Let $G$ be a real classical group of type $B$, $C$, $D$ (including the real metaplectic group). We consider a nilpotent adjoint orbit $check{mathcal O}$ of $check G$, the Langlands dual of $G$ (or the metaplectic dual of $G$ when $G$ is a real metaplectic group). We classify all special unipotent representations of $G$ attached to $check{mathcal O}$, in the sense of Barbasch and Vogan. When $check{mathcal O}$ is of good parity, we construct all such representations of $G$ via the method of theta lifting. As a consequence of the construction and the classification, we conclude that all special unipotent representations of $G$ are unitarizable, as predicted by the Arthur-Barbasch-Vogan conjecture. We also determine precise structure of the associated cycles of special unipotent representations of $G$.
Fixing an arithmetic lattice $Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Delta$ with $[Gamma : Gamma cap Delta] [Delta: Gamma cap Delta] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smiths Subgroups of finite index in nilpotent groups apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $mathbb{Z}$-group with arithmetic lattice $Gamma$, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in $p^{-s}$, where the degrees of the numerator and denominator are independent of $p$. This gives regularity results for the set of arithmetic lattices in $G$.
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