Post on 11-Aug-2020
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).