schubert eisenstein series
TRANSCRIPT
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