Arithmetic Harmonic Analysison holomorphic symplectic quotientsBased on joint work with Letellier and Villegas

arXiv:0810.2076

Tamas Hausel

Royal Society URF at University of Oxfordhttp://www.maths.ox.ac.uk/∼hausel/talks.html

April 2009Mathematics ColloquiumSeoul National University

Frobenius on group characters

I Frobenius, F.G.: Uber Gruppencharacktere (1896):

” I shall develop the concept [of character for arbitrary finitegroups] here in the belief that through its introduction, grouptheory will be substantially enriched. ”

I After proving the orthogonality relations (k = 2 below)Frobenius’ first application was:

Theorem (Frobenius 1896)

Let C1, . . . , Ck ⊂ Γ be conjugacy classes in a finite group Γ then

#{a1 ∈ C1, . . . , ak ∈ Ck |a1a2 · · · ak = 1} =

=∑

χ∈Irr(Γ)

χ(1)2

|Γ|

k∏i=1

χ(Ci )|Ci |χ(1)

Harmonic analysis on finite groups

I Γ finite group; the convolution of f1, . . . , fk : Γ→ C is defined:

f1 ? f2 ? · · · ? fk(h) =∑

g1,g2,...,gk∈Γg1g2...gk=h

f1(g1)f2(g2) . . . fk(gk)

I Fourier transform:

f : C(Γ)→ C ; f : Irr(Γ)→ C f : Irr(Γ)→ C ;ˆf : C(Γ)→ C

f (X ) =∑C∈C(Γ)

X (C)f (C)|C|X (1)

ˆf (C) =∑

χ∈Irr(Γ) χ(C)f (χ)χ(1)

I Fourier inversion formula: ˆf (h) = |Γ|f (h−1)

Fourier of convolution: f1 ? f2 = f1 · f2

I #{a1 ∈ C1, . . . , ak ∈ Ck |a1a2 · · · ak = 1} = 1C1 ? · · · ? 1Ck (1) =

1|Γ|

1C1 · · · 1Ck (1) =∑

χ∈Irr(Γ)χ(1)2

|Γ|∏k

i=1χ(Ci )|Ci |χ(1)

Arithmetic harmonic analysis for GLn(Fq)

I Γ = GLn(Fq)

I character table of GLn(Fq) was calculated by(Jordan, Schur, 1907) for n = 2(Steinberg, 1951) for n = 3, 4(Green, 1955) for all n

I C′1, . . . , C′k ⊂ GLn(Fq) generic semisimple(Hausel, Letellier, Villegas, 2008) calculated explicitly∑

χ∈Irr(GLn(Fq))χ(1)2

|GLn(Fq)|∏k

i=1χ(C′

i )|C′i |

χ(1) =

#{a1 ∈ C′1, . . . , ak ∈ C′k |a1a2 · · · ak = 1} == |Hom(π1(P1

C),GL(n,Fq))| = |PGL(n,Fq)|#{MB(Fq)},

C1, . . . , Ck ⊂ GL(n,C) generic semisimple, Ci (Fq) = C′i

MB = {(A1, . . . ,Ak) Ai ∈ Ci |A1 · · ·Ak = Id}//GL(n,C)

character variety of P1C , as a group-valued symplectic quotient

Example

n = 3, C′i ⊂ GL3(Fq) regular semisimple:

#{MB(Fq)} =∑

χ∈Irr(GL3(Fq))

χ(1)2

|GL3(Fq)|

k∏i=1

χ(C′i )|C′i |χ(1)

=

=

((q + 1)

(q2 + q + 1

))k(q3 − 1)2 (q2 − 1)2

−(3 q2 (q + 1)

)kq4 (q2 − 1)2 (q − 1)2

+ 1/3

(6 q3

)kq6 (q − 1)4

+

(2 q2

(q2 + q + 1

))kq4 (q3 − 1)2 (q − 1)2

+

(q3 (q + 1)

(q2 + q + 1

))kq6 (q3 − 1)2 (q2 − 1)2

−(3 q3 (q + 1)

)kq6 (q2 − 1)2 (q − 1)2

.

e.g. k = 3 ; #{MB(Fq)} = q2 + 6q + 1

Mixed Hodge Structure of Deligne

I⊕

p,q Hp,q;k(M) is the associated graded to the weight and

Hodge filtrations on the cohomology Hk(M,C) of a complexalgebraic variety M

I hp,q;k = dim(Hp,q;k(M)), mixed Hodge numbers

I H(M; x , y , t) =∑p,q,k

hp,q;k(M)xpyqtk , mixed Hodge

polynomial

I P(M; t) = H(M; 1, 1, t), Poincare polynomial

I E (M; x , y) = xnynH(1/x , 1/y ,−1), E-polynomial of a smoothvariety M.

Arithmetic and topological content of the E-polynomial

Theorem (Katz 2006)

If M is a smooth quasi-projective variety defined over Z and

#{M(Fq)} = E (q)

is a polynomial in q, then E (M; x , y) = E (xy).

I MHS on H∗(M,C) is pure if hp,q;k = 0 unless p + q = k ⇔H(M; x , y , t) = (xyt2)nE (−1

xt ,−1yt ) ⇒

P(M; t) = H(M; 1, 1, t) = t2nE (−1t ,−1t );

examples: smooth projective varieties, MDol, MDR,Nakajima’s quiver varieties

I in general the pure part of H(M; x , y , t) isPH(M; x , y) = CoeffT 0

(H(M; xT , yT , tT−1)

); which, for a

smooth M, is always the image of the cohomology of asmooth compactification

Example

n = 3, Ci regular semisimple (q = xy):

E (MB, q) =((q + 1)

(q2 + q + 1

))k(q3 − 1)2 (q2 − 1)2

−(3 q2 (q + 1)

)kq4 (q2 − 1)2 (q − 1)2

+ 1/3

(6 q3

)kq6 (q − 1)4

+

(2 q2

(q2 + q + 1

))kq4 (q3 − 1)2 (q − 1)2

+

(q3 (q + 1)

(q2 + q + 1

))kq6 (q3 − 1)2 (q2 − 1)2

−(3 q3 (q + 1)

)kq6 (q2 − 1)2 (q − 1)2

.

e.g. k = 3 ; E (MB; q) = q2 + 6q + 1,

dim(MB) = 2 and MB affine ⇒ MHS is not pure on H∗(MB)

What is H(MB; q, t)? What is PH(MB; q)?

Conjecture (Hausel, 2004)

When n = 3, Ci regular semisimple, hp,p;d = hN−p,N−p;d+N−2p for

H(MB, q, t) =∑

hp,p;dqptd =

((qt2 + 1

) (q2t4 + qt2 + 1

))k(q3t6 − 1) (q3t4 − 1) (q2t4 − 1) (q2t2 − 1)

−(3 q2t4

(qt2 + 1

))kq4t8 (q2t4 − 1) (q2t2 − 1) (qt2 − 1) (q − 1)

+ 1/36k(qt2)3 k

q6t12 (qt2 − 1)2 (q − 1)2

+

(q2t4

(2 q2t2 + qt2 + q + 2

))kq4t8 (q3t4 − 1) (q3t2 − 1) (qt2 − 1) (q − 1)

+

(q3t6 (q + 1)

(q2 + q + 1

))kq6t12 (q3t2 − 1) (q3 − 1) (q2t2 − 1) (q2 − 1)

−(3 q3t6 (q + 1)

)kq6t12 (q2t2 − 1) (q2 − 1) (qt2 − 1) (q − 1)

,

e.g. k = 3 ; H(MB; q, t) = 1 + 6qt2 + q2t2

The Riemann-Hilbert map

I g = glnI C = P1 with punctures a1, . . . , ak ∈ P1

I Ci semisimple adjoint orbit in g(C)

I Q = {(A1, . . . ,Ak) Ai ∈ Ci |A1 + · · ·+ Ak = 0}//G (C),star-shaped quiver variety, as holomorphic symplectic quotient

I Q is smooth when Ci are generic

I “Q ⊂MDR”, a point in Q gives the meromorphic flatGL(n,C)-connection

∑Ai

dzz−ai

on the trivial bundle on C .

I Ci = exp(2πiCi ) ⊂ G (C) is the corresponding conjugacy class

I the Riemann-Hilbert monodromy map

νa : Q →MB

is given by sending the flat connection to its holonomy.

The purity conjecture

Conjecture (Hausel, 2004)

If Ci are generic, then ν∗a : PH∗(MB)∼=−→ H∗(Q)

Example

I n = 3, k = 3, Ci regular semisimple

I Q is E6 ALE space,

I MB∼=MDol elliptic fibration with singular fibre of type E6.

I P(Q; t) = 1 + 6t2

I H(MB; q, t) = 1 + 6qt2 + q2t2

⇓

Conjecture is true in this case

Fourier transform for Q

I Ψ : Fq → C non-trivial additive character

I f : g(Fq)→ C its Fourier transform f : g∗(Fq)→ C

f (Y ) :=∑

X∈g(Fq)

f (X )Ψ(〈X ,Y 〉).

I 1Ci : g(Fq)→ C characteristic function of Ci ⊂ g(Fq)

Theorem

|PGL(n,Fq)|#{Q(Fq)} =#{a1 ∈ C1, . . . , ak ∈ Ck |a1 + a2 + · · ·+ ak = 0} =

1C1 ? · · · ? 1Ck (0) = 1C1 · · · 1Ck (0) =1

|gln(Fq)|∑

X∈g(Fq) 1C1(X ) · · · · · 1Ck (X )

TheoremLetellier’s character table for gl3(Fq) implies that when n = 3 andall adjoint orbits Ci are regular semi-simple

P(Q; t) =

((t2 + 1

) (t4 + t2 + 1

))k(t6 − 1) (t4 − 1)

−(3 t4

(t2 + 1

))kt8 (t4 − 1) (t2 − 1)

+ 1/36k(t2)3 k

t12 (t2 − 1)2

−(t4(t2 + 2

))kt8 (t2 − 1)

+ t6k−12 +(3 t6)k

t12 (t2 − 1)

which agrees with the pure part of the conjectured mixed Hodgepolynomial of the corresponding MB.

Master Conjecture

Conjecture (Hausel-Letellier-Villegas, 2008)

µ = (µi )ki=1 ∈ P(n){1,...,k} type of the conjugacy classes (Ci )k

i=1∑p,k hp,p;k(Mµ

B)qptk = (t√

q)dµ(q − 1)(1− 1qt2 ) ·

·⟨

Log(∑

λ∈P

(∏ki=1 Hλ(xi ; q, 1

qt2 ))Hλ(q, 1

qt2 )), hµ⟩ ,

where Hλ(xi ; q, 1qt2 ) are the Macdonald symmetric functions.

Theorem (Hausel-Letellier-Villegas, 2008)

I The Master Conjecture is true when specialized to t = −1giving a formula for E (MB; q) = #{MB(Fq)}.

I Taking the pure part of the Master Conjecture gives thePoincare polynomial of the quiver variety Q, consistently withthe purity conjecture.

I When k = 2 the Master Conjecture is true and reduces to theCauchy identity for Macdonald polynomials; thus it is adeformation of Frobenius’ orthogonality for GLn(Fq).