Schubert Eisenstein Series

YoungJu Choie

Dept of Math

Pohng Mathematical Institute

POSTECH,Pohang, Korea

Talk at ICERM

Jan 30, 2013

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 1 / 12

Where is POSTECH(Pohang university of Science and Technology)?

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 2 / 12

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 3 / 12

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 4 / 12

그림: POSTECH (1986- ) http://postech.ac.kr

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 5 / 12

What is ”Schubert Eisenstein” series?

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 6 / 12

What is Schubert Eisenstein Series?

Schubert Eisenstein series is defined as sums like usual Eisenstein series

but with the summation restricted to elements coming from a particular

Schubert cell.

Let G be a split semisimple algebraic group over a global field F and B be

its Borel subgroup.

The usual Eisenstein series are sums over B(F )\G (F ), that is, over the

integer points in the flag variety X = B\G .Given a Weyl group element w , one may consider the sum restricted to a

single Schubert cell Xw . This is called a Schubert Eisenstein series Ew .

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 7 / 12

Schubert Cell

More precisely, consider the Bruhat decomposition of G

G =⋃

w∈WBwB

where W is the Weyl group.

This gives the decomposition of the flag variety into Schubert cells

X = ∪w∈WYw

where Yw is the image of BwB in X = B\G .The Schubert cell Xw is the Zariski closure of Yw :

Xw :=⋃

u ∈W , u ≤ w

Yu,

where u ≤ w is the Bruhat order.

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 8 / 12

Schubert Eisenstein Series

Define the Schubert Eisenstein series

Ew (g , ν) =∑

γ∈Xw (Z)

fν(γg)

where

fν(bg) = (δ1/2χν)(b) f (g), b ∈ B(A).

a character χν on T (A)/T (F ), ν ∈ T̂ and δ is a modular quasicharacter.

If w0 is the long Weyl group element, Ew0(g , ν) is the usual Eisenstein

series , so automorphic object.

However, in general Schubert Eisenstein series is no longer automorphic!

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 9 / 12

We would like to explore...

· Does SE have meromorphic continuation to all values of the

parameters?

· Do they have some functional equations?

· One may represent SE recursively using Bott-Samelson map if

Bott-Samelson variety is isomorphic to Schubert variety. How to

represent SE when Bott-Samelson map is not isomorphic?

· Is there any arithmetic implication?

· ... ? more connections with others?

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 10 / 12

Affermative answers by explicit computation

in the case when G = GL(3) with Bump

그림: D. Bump (Stanford U)

Now working on G = GL(4) case with Bump ,

More..YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 11 / 12

YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 12 / 12