数理解析研究所講究録
第 1874 巻 2014 年 94-97

94

Schubert Eisenstein Series for $GL(3)$Daniel Bump and YoungJu Choie

1Schubert Eisenstein series

We try to explain the concept of Schubert Eisenstein series and possible arithemetic
implication. This is joint work with D. Bump and a survey talk at RIMS. The
original work can be found in [1]1.1

Schubert Eisenstein series

Let us start with a general set up.
Let
be a split reductive algebraic group over a global field . Let
be thewith opposite root data, so thatis the conmaximal torus of the group. Then parametrizes a character
nected Langlands -group. Letof$T(A)/T(F)$ , where A is the adele ring of . Extending
to the Borel subgroup$B(A)$ , letbe an element of the corresponding induced representation, so that
$F$

$G$

$\hat{G}$$L$$\hat{T}$$\hat{G}(\mathbb{C})$$\nu\in\hat{T}(\mathbb{C})$$v$$\chi_{\nu}$$F$$\chi_{\nu}$$f_{\nu}$$f_{\nu}(bg)=(\delta^{1/2}\chi_{\nu})(b)f(g) , b\in B(A)$.

(1)

The flag variety $X=B\backslash G$ is a projective variety. We recall its decomposition
into Schubert cells. We have the Bruhat decomposition $G=\cup BwB$ , a disjoint
union over $w\in W$ , and let
be the image of $BwB$ in $X$ . The Schubert cellisthe Zariski closure of . It equals$X_{w}$$Y_{w}$$Y_{w}$$u\in W\bigcup_{u\leq w}Y_{u},$be the subset of that is the union of $BuB$
where $u\leq w$ is the Bruhat order. Let
belonging tofor $u\leq w$ . It is not a subgroup. Let $X_{w}(F)$ be the set of. Thus $X_{w}(F)=B_{F}\backslash G_{w}(F)$ . We may now define the Schubert Eisenstein series$G$$G_{w}$$\gamma\in B_{F}\backslash G_{F}$$X_{w}$$E_{w}(g, f, \chi)=\sum_{\gamma\in X_{w}(F)}f(\gamma g)$.

95
1.2

Bott-Samelsom varieties

Let us recall the Bott-Samelson varieties and their relationship with Schubert varieties. We will denote by and the simple roots and corresponding simple reflections. Let $w\in W$ and let
be a reduced decomposition of. Let be the minimal parabolic
into a product of simple reflections:byon
subgroup generated by and . We define a left action of$s_{i}$$\alpha_{i}$$w$$\mathfrak{w}=(s_{i_{1}}, s_{i_{2}}, \cdots, s_{i_{k}})$$P_{j}$$w=s_{i_{1}}\cdots s_{i_{k}}$$B^{k}$$B$$s_{j}$$P_{i_{1}}\cross\cdots P_{i_{k}}$$(b_{1}, \cdots, b_{k})\cdot(p_{i_{1}}, \cdots,p_{i_{k}})=(b_{1}p_{i_{1}}b_{2}^{-1}, b_{2}p_{i_{2}}b_{3}^{-1}, \cdots, b_{k}p_{i_{k}})$The quotientmorphism:$Z_{t\mathfrak{v}}arrow X_{w}$(2). There is a
is the Bott-Samelson variety
induced by the multiplication map that sends$B^{k}\backslash (P_{i_{1}}\cross\cdots\cross P_{i_{k}})$$BS\mathfrak{w}$.

$Z_{\mathfrak{w}}$$(p_{i_{1}}, \cdots,p_{i_{k}})\mapsto p_{i_{1}}\cdots p_{i_{k}}.$This map is a surjective birational morphism.
It is known that Bott-Samelson varieties are always nonsingular, so this gives a:. The map
resolution of the singularities of the Schubert variety
may not be an isomorphism. In special cases where it is an isomorphism, every, where if
element of
has a unique representation as a productis a $ro$ ot (in this case a simple root)is the Chevalley embedding of $SL(2)$ into
corresponding to , so the image oflies in the Levi subgroup of . Beyond
these special cases whereis an isomorphism, in every case each element of
has such a factorization, and if the element is in general position, it is unique, sinceis birational. Let us call this a Bott-Samelson factorization. This means that
we may write$X_{w}$$X_{w}$$BS\mathfrak{w}$$Z_{\mathfrak{w}}arrow X_{w}$$i_{\alpha 1}(\gamma_{1})\cdots\iota_{\alpha}k(\gamma_{k})$$\alpha$$\iota_{\alpha}$$G$$P_{i}$$\alpha$$\iota_{\alpha i}$$X_{w}$$BS\mathfrak{w}$$BS_{t\mathfrak{v}}$$E_{s1s_{k}}(g, v)= \sum_{k\gamma\in B_{SL_{2}}(F)\backslash SL_{2}(F)}E_{S1s_{k-1}}(\iota_{\alpha}k(\gamma_{k})g, v)$(3),:building up the Schubert Eisenstein series by repeatedsummations. Ifis not an isomorphism, a modification of this method should be applica$SL_{2}$$BS_{\mathfrak{w}}$$Z_{\mathfrak{w}}arrow X_{w}$ble.

1.3$GL_{3}$Schubert Eisenstein series, Explicit Computation

Let us be more precise: let $G=GL_{3}$ and let$\zeta^{*}(s)=|D_{F}|^{\frac{\epsilon}{2}}\prod\zeta_{v}(s)$,$\zeta_{v}(s)=\{\begin{array}{ll}(1-T_{v}^{-s})^{-1} if v is nonarchimedean,\Gamma_{v}(s) if v is archimedean\end{array}$

96
is the discriminant of . For simplicity we will assume that
where we recall thatthe character is unramified at every place. Findsuch that$F$$D_{F}$$v_{1},$$\chi$$v_{2}\in \mathbb{C}$$(\delta^{1/2}\chi)(y_{1} y_{2} y_{3})=|y_{1}|^{2_{1}\nu}\nu+2|y_{2}|^{\nu-\nu}21|y_{3}|^{-\nu_{1}-2\nu}2.$

We will denote this character$\chi_{\nu\nu}1,2^{\cdot}$

Also, take $f=f^{o}$ where$f^{o}(g)=f_{\nu 1,\nu 2}^{o}(g)= \prod_{v}f_{v}^{o}(g_{v})$.

Thus if $k\in K$$f_{\nu,\nu}^{o}12((y_{l} y_{2}* **y_{3})k)=|y_{1}|^{2\nu+\nu}12|y_{2}|^{\nu-\nu}21|y_{3}|^{-\nu-2\nu}12.$

For each $w\in W$ normalize the Schubert Eisenstein series and denote$E_{w}^{*}(g;v_{1}, v_{2})=\zeta^{*}(3v_{1})\zeta^{*}(3v_{2})\zeta^{*}(3\nu_{1}+3\nu_{2}-1)E_{w}(g;v_{1}, \nu_{2})$and$\hat{E}_{s}^{*}1^{S}2(\nu_{1}, v_{2})=E_{ss_{2}}^{*}1(\nu_{1}, v_{2})-E_{s_{2}}^{*}(v_{1}, v_{2})-E_{S2}^{*}(\nu_{2},1-\nu_{1}+v_{2})$

.

(4)

.

(5)

Similarly$\hat{E}_{s_{2}s1}^{*}(\nu_{1}, \nu_{2})=E_{s_{2}s1}^{*}(\nu_{1}, v_{2})-E_{S1}^{*}(\nu_{1}, v_{2})-E_{S1}^{*}(1-\nu_{1}+v_{2}, v_{1})$Theorem 1 [$1JE_{S1^{S}2}^{*}(g;\nu_{1}, v_{2})$has meromorphic continuation to allfunctional equation$E_{1^{S}2}^{*}s(g; \nu_{1}, \nu_{2})=E_{S182}^{*}(g;\frac{2}{3}-\nu_{1}, v_{1}+\nu_{2}-\frac{1}{3})$Moreover$\hat{E}_{s}^{**}1^{S}2(g;\nu_{1}, v_{2})$is an entire function.

The similar result holds for$\hat{E}_{ss1}^{*}2(\nu_{1}, \nu_{2})$.

$\nu_{1},$$v_{2}$. It has a97
1.4

Kronecker Limit Formula

Bump and Goldfeld proved the following result. If
is a cubic field, and is anideal class of $K$ one may associate with a compact torus of, and if
is theperiod ofover this torus, then the Taylor expansion of the -function
has the form. Therefore if is a character of the ideal class groupthen. The proof involves showing that the torus period of theEisenstein series equals a Rankin-Selberg integral of a Hilbert modular Eisensteinseries.
$K/\mathbb{Q}$$\mathfrak{a}$$GL_{3}$$\mathfrak{a}$$\kappa(g)$$L_{\mathfrak{a}}$$L$$\rho s^{-1}+L_{\mathfrak{a}}+\cdots$$L(s, \mathfrak{a})$$\theta$$L(s, \theta)=\sum\theta(\mathfrak{a})L_{a}$Considering Taylor expansions of$E_{w}$

for various$w$

at $\nu_{1}=v_{2}=0$ we get

Theorem 2 [lJ We have$\kappa(g)=\frac{\rho}{3}\zeta^{*}(2)[\hat{E}_{s_{2^{S}1}}^{**}(g;0,0)+E_{s_{1}}^{**}(g;1,0)]+c_{0}.$Furthermore$\kappa(g)=\frac{\rho}{3}\zeta^{*}(2)[\hat{E}_{1}^{**}ss_{2}(g;1,0)+\phi_{S_{2}}(g)]+c_{0}’.$Acknowledgement I would like to thank to Prof Koji Chinen, the organizer ofthe RIMS conference on analytic number theory 2012 for his kind invitation. My
special thanks goes to Prof Kohji Matsumoto for his kind invitation, arrangement
and hospitality while I was in Nagoya and Kyoto.
This work is supported by partially supported by NRF 2012047640, NRF 2011-0008928 and NRF 2008-0061325

Address: D. Bump, Dept of Mathematics, Stanford University, $CA$ , USA
Address: Y. Choie, Dept of Mathematics, POSTECH, Pohang, Korea

References[1] Daniel Bump and YoungJu Choie, Schubert Eisenstein Series for $GL(3)$ ,Preprint (2011)