On the Kuznetsov Trace Formula for $\mathrm{PGL}_2(\mathbb{C})$
Zhi Qi
备注：
19 pages. J. Funct. Anal. Some calculations and statements for complementary series are simplified (corrected), in which case d should be 0 (not an arbitrary integer)
In this note, using a representation theoretic method of Cogdell and Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary discrete group $Γ$ in $\mathrm{PGL}_2(\mathbb{C})$ that is cofinite but not cocompact. An essential ingredient is a kernel formula, recently proved by the author, on Bessel functions for $\mathrm{PGL}_2(\mathbb{C})$. This approach avoids the difficult analysis in the existing method due to Bruggeman and Motohashi.
We determine the Jordan constants of groups $\mathrm{GL}_2(K)$, $\mathrm{SL}_2(K)$, $\mathrm{PGL}_2(K)$ and $\mathrm{PGL}_3(K)$ for any given field $K$ of characteristic 0.
We relate the analytic conductor of a generic irreducible representation of $\mathrm{GL}_n(\mathbb{R})$ to the invariance properties of vectors in that representation. The relationship is an analytic archimedean analogue of some aspects of the classical non-archimedean newvector theory of Casselman and Jacquet--Piatetski-Shapiro--Shalika. We illustrate how this relationship may be applied in trace formulas to majorize sums over automorphic forms on $\mathrm{PGL}_n(\mathbb{Z}) \backslash \mathrm{PGL}_n(\mathbb{R})$ ordered by analytic conductor.
Pappus Theorem, Schwartz Representations and Anosov Representations
在“Pappus定理与模群“一文中，R. Schwartz通过Pappus定理将模群$\mathrm{PSL}(2,\mathbb{Z})$的2维忠实表示族$ρ_Θ$构造成射影平面的射影对称群$\mathscr{G}$。$\mathrm{PSL}(2,\mathbb{Z})$的唯一索引2子群$\mathrm{PSL}(2,\mathbb{Z})_o$在每个表示$ρ_Θ$下的像在$\mathscr{G}$的子群$\mathrm{PGL}(3,\mathbb{R})$中，并且在标志簇中保留拓扑圆，但是$ρ_Θ$不是Anosov。在她的博士论文中，V. P. Valério阐明了Schwartz表示的Anosov样特征：对于每个$ρ_Θ$，存在一个1维的Anosov表示族$ρ^\varepsilon_Θ$，它将$\mathrm{PSL}(2,\mathbb{Z})_o$转化为$\mathrm{PGL}(3,\mathbb{R})$，其极限是将$ρ_Θ$限制为$\mathrm{PSL}(2,\mathbb{Z})_o$。在本文中，我们改进了她的工作：对于每个$ρ_Θ$，我们建立了一个二维家庭的Anosov表示的$\mathrm{PSL}(2,\mathbb{Z})_o$到$\mathrm{PGL}(3,\mathbb{R})$包含$ρ^\varepsilon_Θ$和一个1维的子类，其中可以…
Let $G$ be a smooth connected reductive group over a field $k$ and $Γ$ be a central subgroup of $G$. We construct Eilenberg-Moore-type spectral sequences converging to the Hodge and de Rham cohomology of $B(G/Γ)$. As an application, building upon work of Toda and using Totaro's inequality, we show that for all $m\geq 0$ the Hodge and de Rham cohomology algebras of the classifying stacks $B\mathrm{PGL}_{4m+2}$ and $B\mathrm{PSO}_{4m+2}$ over $\mathbb{F}_2$ are isomorphic to the singular $\mathbb{F}_2$-cohomology of the classifying space of the corresponding Lie group. From this we obtain a full…
For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the $G$-character variety of a compact Riemann surface $C$ and of the moduli space of $G$-Higgs bundles on $C$ of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.
Let $q$ be a prime power and $\mathbb{F}_q$ be the finite field with $q$ elements. In this article we investigate the space of unramified automorphic forms for $\mathrm{PGL}_n$ over the rational function field defined over $\mathbb{F}_q$ (i.e.\ for $\mathbb{P}^1$ defined over $\mathbb{F}_q$). In particular, we prove that the space of unramified cusp form is trivial and (for $n=3$) that the space of eigenforms is one dimensional. Moreover, we show that there are no nontrivial unramified toroidal forms for $\mathrm{PGL}_3$ over $\mathbb{P}^1$ and conjecture that the space of all toroidal automor…
In this paper, we carry out several computations involving graded (or $\mathbb{G}_{\mathrm{m}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the $\mathbb{G}_{\mathrm{m}}$-action on certain normalized (or "canonical") simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the coho…
Let $G$ be a finite reductive group defined over $\mathbb{F}_q$, with $q$ a power of a prime $p$. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of $G$ into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants of the endomorphism algebra of a Gelfand-Graev representation of $G$ when $G=\mathrm{PGL}_3(q)$ for an arbitrary prime $p$, and when $G=\mathrm{SO}_5(q)$ for $p$ odd.
