My Works on Non-Abelian Zetas
Since 2009
\[\huge{\color{Brown}{\mathbb{\ Rank\ n\ Non-Abelian\ Zeta\ Functions\ }}}\]
\[\]
\(\bigodot\ \Large{\mathcal M_{\mathbb Q,n}\ \mathrm{denotes\ the\ moduli\ space\ of\ semi\!\!-\!\!stable\ lattices\ \Lambda\ of\ rank}\ n\ in\ \mathbb R^n}\)
\[\]
\(\bigodot\ \Large{h^0(\mathbb Q,\Lambda), resp. \mathrm{deg}(\Lambda),\ \mathrm{denotes\ the\ 0\!\!-\!\!th\ arithmetic\ cohomology,\ resp.\ the\ arithmetic\ degree,\ of\ the\ lattice}\ \Lambda}\)
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Rank\ n\ Zeta\ Function\ for\ \mathbb Q:}}}\)
\[\boxed{\begin{align*}\huge{\widehat\zeta_{\mathbb Q,n}(s):=}&\huge\int_{\mathcal M_{\mathbb Q,n}}\big(e^{h^0(\mathbb Q,\Lambda)}-1\big)\big(e^{-s}\big)^{\mathrm{deg}(\Lambda)}d\mu(\Lambda)\ \ \Re(s)>1\end{align*}}\]
\[\]
\(\huge{\spadesuit\ \color{Blue}{\mathrm{Relation\ with\ Abelian\ Zeta\ Function:}}}\)
\[\boxed{\huge{
\widehat\zeta_{\mathbb Q,1}(s)=\widehat\zeta(s).}}\]
\(\huge{\spadesuit\ \color{Blue}{\mathrm{Functional\ Equation:}}}\)
\[\]
\[\boxed{\huge{
\widehat\zeta_{\mathbb Q,n}(1-s)=\widehat\zeta_{\mathbb Q,n}(s)}}\]
\[\]
\(\huge{\spadesuit\ \color{Blue}{\mathrm{Geometric Interpretation of the Residue:}}}\)
\(\huge{\widehat\zeta_{\mathbb Q,n}(s)\ \mathrm{admits\ only\ two\ singularities,\ namely,\ two\ simple\ poles\ at\ s=0,1.\ Moreover,}}\)
\[\boxed{\huge{
\mathrm{Res}_{s=1}\Big(\widehat\zeta_{\mathbb Q,n}(s)\Big)=\mathrm{Vol}\Big(\mathcal M_{\mathbb Q,n}[1]\Big)}}\]
\[\]
\(\huge{\spadesuit\ \color{Blue}{\mathrm{Riemann\ Hypothesis:}}}\)
\[\boxed{\huge{\widehat\zeta_{\mathbb Q,n}(s)=0\qquad\Rightarrow\qquad\mathrm{Re}(s)=\frac{1}{2}
}}\]
(A weak version of this is proved as Theorem 15.4 of my book on Zeta Functions of Reductive Groups and Their Zeros published by the World Scientific, 2018)
\[\]
\[\]
\[\huge{\color{Brown}{\mathbb{\ Zeta\ Functions\ For\ (G,P)/Q}}}\]
\[\]
\(\bigodot\ \Large{G:\ \mathrm{a\ reductive\ group\ with\ P\ its\ maximal\ parabolic\ subgroup,\ all\
defined\ over}\ \mathbb Q
}\)
\[\]
\(\bigodot\ \Large{\Phi^\pm:\ \mathrm{the\ associated\ system\ of\ positive,\ resp. negative, roots}
}\)
\[\]
\(\bigodot\ \Large{\Delta:\ \mathrm{the\ set\ of\ simple\ roots}
}\)
\[\]
\(\bigodot\ \Large{W:\ \mathrm{the\ Weyl\ group}
}\)
\[\]
\(\bigodot\ \Large{\rho\,:=\,\frac{1}{2}\sum_{\alpha\in\Phi^+}\alpha\ \mathrm{the\ Weyl\ vector}
}\)
\[\]
\(\bigodot\ \Large{\{\beta_1,\beta_2,\dots,\beta_{|G|-1}\}:=\Delta-\{\alpha_P\}
\ \mathrm{with}\ \alpha_P\ \mathrm{the\ simple\ root\ corresponding\ to}\ P
}\)
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Weng\ Zeta\ Functions\ for\ (G,P)}/\mathbb Q:}}\)
\[\boxed{\begin{align*}&\Large{\widehat\zeta^{G/P}_{\mathbb Q}(s):=}\\
&:=\Large{\mathrm{Norm}}\,\Bigg({\Large\mathrm{Res}}_{\Large{\langle \lambda-\rho,\beta_1^\vee\rangle=0,\,\langle \lambda-\rho,\beta_2^\vee\rangle=0,\, \dots,\, \langle \lambda-\rho,\beta_{|G|-1}^\vee\rangle=0}}\,\Big(
{\Large\sum}_{w\in W}\frac{\Large{e^{-\langle \lambda,T\rangle}}}{{\huge\prod}_{\alpha\in\Delta}\langle \lambda-\rho,\alpha^\vee\rangle}\cdot{\Large\prod}_{\alpha\in \Phi^+, w\alpha\in\Phi^-}
\frac{\widehat\zeta\big(\langle\lambda,\alpha^\vee\rangle\big)}{\widehat\zeta\big(\langle\lambda,\alpha^\vee\rangle+1\big)}\Big)\Bigg)
\end{align*}}\]
\[\]
\[\]\(\huge{\spadesuit\ \color{Blue}{\mathrm{Relation\ with\ Non\!\!-\!\!Abelian\ Zeta\ Function:}}}\)
\[\]
\(\huge{\mathrm{Up\ to\ a\ constant\ factor\ and\ an\ affine\ change\ of\ variables,}}\)
\[\boxed{\huge{\widehat\zeta_{\mathbb Q}^{{\mathrm SL}_n/P_{n-1,1}}(s)=\widehat\zeta_{\mathbb Q,n}(s).}}\]
\[\]
\[\]
\(\huge{\spadesuit\ \color{Blue}{\mathrm{Functional\ Equation:}}\qquad \exists\qquad {constant}\quad c_{G/P},\quad \mathrm{s.t.}}\)
\[\boxed{\huge{
\widehat\zeta^{G/P}_{\mathbb Q}(c_{P/Q}-s)=\widehat\zeta^{G/P}_{\mathbb Q}(s)}}\]
\[\]
\[\]
\(\huge{\spadesuit\ \color{Blue}{\mathrm{Riemann\ Hypothesis:}}}\)
\[\boxed{\huge{\widehat\zeta^{G/P}_{\mathbb Q}(s)=0\qquad\Rightarrow\qquad\mathrm{Re}(s)=\frac{c_{G/P}}{2}
}}\]
(Existence of a Functional equation was conjectured by L.Weng and proved by Y. Komori. A weak version of the Riemann Hypothesis is proved as Theorem 17.2 of my book on Zeta Functions of Reductive Groups and Their Zeros published by the World Scientific, 2018)
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