The trace formula and prehomogeneous vector spaces
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The trace formula and prehomogeneous vector spaces Werner Hoffmann Bielefeld University, SFB 701
Simons Symposium on Families of Automorphic Forms and the Trace Fromula January 29, 2014
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Induction of conjugacy classes Consider connected linear algebraic groups over a field F and geometric conjugacy classes. Theorem (Lusztig-Spaltenstein for unipotent classes, H.) Let P be a parabolic subgroup of a reductive group G with Levi decomposition P = MN, and C a conjugacy class in M. Then there is a unique dense P-conjugacy class C ′ in CN and a ˜ in G such that C ˜ ∩ P = C ′. unique conjugacy class C S P P ′ ˜ = IndG Notation C P C , C = InflM C , Pinfl = C InflM C
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Induction of conjugacy classes Consider connected linear algebraic groups over a field F and geometric conjugacy classes. Theorem (Lusztig-Spaltenstein for unipotent classes, H.) Let P be a parabolic subgroup of a reductive group G with Levi decomposition P = MN, and C a conjugacy class in M. Then there is a unique dense P-conjugacy class C ′ in CN and a ˜ in G such that C ˜ ∩ P = C ′. unique conjugacy class C S P P ′ ˜ = IndG Notation C P C , C = InflM C , Pinfl = C InflM C Theorem (Lusztig-Spaltenstein, H.)
G If M is a Levi component of both P and Q, then IndG P C = IndQ C . Given γ ∈ G , there are only finitely many P such that γ ∈ Pinfl .
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Prehomogeneous varieties Definition • A prehomogeneous G -variety is an irreducible variety V with an algebraic G -action such that there exists a dense G -orbit O. • It is called special if every relatively G -invariant rational function (defined over any extension of F ) is constant. • (V , G ) is called a prehomogeneous vector space if V is a vector space and the G -action is linear. • It is called regular if V ∗ is prehomogeneous and dp/p : O → V ∗ is a dominant morphism for some relative invariant p. p(gx) = χ(g )p(x).
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Prehomogeneous varieties Definition • A prehomogeneous G -variety is an irreducible variety V with an algebraic G -action such that there exists a dense G -orbit O. • It is called special if every relatively G -invariant rational function (defined over any extension of F ) is constant. • (V , G ) is called a prehomogeneous vector space if V is a vector space and the G -action is linear. • It is called regular if V ∗ is prehomogeneous and dp/p : O → V ∗ is a dominant morphism for some relative invariant p. p(gx) = χ(g )p(x). Zeta integral (F number field, A ring of adeles): Z X φ(g ξ) dg |χ1 (g )|s1 . . . |χr (g )|sr Z (φ, s1 , . . . , sr ) = G (A)/G (F )
ξ∈O(F )
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Canonical parabolic subgroups G reductive, char F = 0. Jacobson-Morozov: ∀ X ∈ g nilpotent, ∃ H, Y ∈ g such that [H, X ] = 2X ,
[H, Y ] = −2Y ,
[X , Y ] = H.
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Canonical parabolic subgroups G reductive, char F = 0. Jacobson-Morozov: ∀ X ∈ g nilpotent, ∃ H, Y ∈ g such that [H, Y ] = −2Y ,
[H, X ] = 2X ,
[X , Y ] = H.
Canonical parabolic of X and of exp X ∈ G (F ): M q= gn , Q = NormG q, n≥0
where gn = {Z ∈ g | [H, Z ] = nZ }.
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Canonical parabolic subgroups G reductive, char F = 0. Jacobson-Morozov: ∀ X ∈ g nilpotent, ∃ H, Y ∈ g such that [H, Y ] = −2Y ,
[H, X ] = 2X ,
[X , Y ] = H.
Canonical parabolic of X and of exp X ∈ G (F ): M q= gn , Q = NormG q, n≥0
where gn = {Z ∈ g | [H, Z ] = nZ }. Theorem (Vinberg, H.) L L Let u = n≥2 gn , u′ = [u, u] = n>2 gn . Then u/u′ is a regular prehomogeneous vector space under L = CentG H.
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Mean value formula Theorem (Siegel/Weil/Ono) If O is a special G -homogeneous variety over a number field F with trivial π1 (O(C)), π2 (O(C)) and X (G ), and if [G (A)ξ ∩ O(F ) : G (F )] is constant on O(F ), then Z Z X φ(g ξ) dg = φ(x) dx. G (A)/G (F ) ξ∈O(F )
G (A)O(F )
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Mean value formula Theorem (Siegel/Weil/Ono) If O is a special G -homogeneous variety over a number field F with trivial π1 (O(C)), π2 (O(C)) and X (G ), and if [G (A)ξ ∩ O(F ) : G (F )] is constant on O(F ), then Z Z X φ(g ξ) dg = φ(x) dx. G (A)/G (F ) ξ∈O(F )
G (A)O(F )
Conjecture There is a normal unipotent subgroup N C of P such that, for γ ∈ C ′ = InflP C , all elements of γN C ∩ C ′ have the same canonical parabolic and γN/N C is special under PγN .
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The geometric side of the trace formula Always write G for G (F ) and G for G (A). Exception: K a maximal compact subgroup of G such that G (F∞ )K is open and G = PK for every parabolic subgroup P. G1 the maximal closed normal subgroup with torsion-free abelian quotient.
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The geometric side of the trace formula Always write G for G (F ) and G for G (A). Exception: K a maximal compact subgroup of G such that G (F∞ )K is open and G = PK for every parabolic subgroup P. G1 the maximal closed normal subgroup with torsion-free abelian quotient. Representation RP of G1 in L2 (NP\G1 ): Z KP (x, y )φ(y ) dy RP (f )φ (x) = G \G1
for f ∈ Cc∞ (G1 ), φ ∈ L2 (NP\G1 ), where X Z f (x −1 γny ) dn. KP (x, y ) = γ∈P/N
N
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Truncation parameter TP ∈ aP = Lie(P/P1 ) Modular character ∆P : P → R+
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Truncation parameter TP ∈ aP = Lie(P/P1 ) Modular character ∆P : P → R+ For P maximal: Let −ˆ τPT be the characteristic function of {p ∈ P | ∆P (p) > ∆P (exp TP )}K For general P, τˆPT (x) =
Y
P ′ ⊃P P ′ max.
τˆPT′ (x)
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Truncation parameter TP ∈ aP = Lie(P/P1 ) Modular character ∆P : P → R+ For P maximal: Let −ˆ τPT be the characteristic function of {p ∈ P | ∆P (p) > ∆P (exp TP )}K For general P, τˆPT (x) =
Y
τˆPT′ (x)
P ′ ⊃P P ′ max.
Compatible family T = (TP )P : T ˆPT (x). τˆγPγ −1 (γx) = τ
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Arthur’s trace distribution: Z J T (f ) =
G \G1
X P
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KP (x, x)ˆ τPT (x) dx
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Arthur’s trace distribution: Z J T (f ) =
G \G1
X
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KP (x, x)ˆ τPT (x) dx
P
For geometric conjugacy class C in G , define JCT (f ) using X Z f (x −1 γny ) dn. KP,C (x, y ) = γ∈P/N IndG γ=C
N
Conjecture XZ C
hence
G \G1
X KP,C (x, x)ˆ τPT (x) dx < ∞,
J T (f ) =
P
P
C
JCT (f ).
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Reordering the unipotent contribution C a unipotent geometric conjugacy class. Z Z X X Z T JC (f ) = f (x −1 γnn′ x) dn′ dn τPT (x) dx. G \G1 P
γ∈P/N IndG γ=C
N/Nγ
Nγ
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Reordering the unipotent contribution C a unipotent geometric conjugacy class. Z Z X X Z T JC (f ) = f (x −1 γnn′ x) dn′ dn τPT (x) dx. G \G1 P
γ∈P/N IndG γ=C
N/Nγ
Nγ
R P For γ ∈ D with InflP D = D ′ replace γN/ND by (D ′ ∩γN)N D /N D using mean-value formula. Ignoring convergence: Z Z X X X T f (x −1 γn′ x) dn′ τPT (x) dx, JC (f ) = G \G1 P D ′ ⊂C ∩P γ∈D ′ N D /N D
where D = D ′ N/N.
ND
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Collect pairs (P, D ′ ) with given N ′ = N D , collect cosets γN D with given canonical parabolic Q (abbreviated γ ∈ Qcan ): Z XX X XZ T JC (f ) = f (x −1 γn′ x) dn′ τPT (x) dx. G \G1 Q N ′ ⊂Q γ∈(C ∩Q )N ′ /N ′ can
N′ P γ∈Pinfl N γ =N ′
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Collect pairs (P, D ′ ) with given N ′ = N D , collect cosets γN D with given canonical parabolic Q (abbreviated γ ∈ Qcan ): Z XX X XZ T JC (f ) = f (x −1 γn′ x) dn′ τPT (x) dx. G \G1 Q N ′ ⊂Q γ∈(C ∩Q )N ′ /N ′ can
N′ P γ∈Pinfl N γ =N ′
Fix a canonical parabolic Q for C : Z Z X X T JC (f ) = Q\G1
N ′ ⊂Q γ∈(C ∩Qcan )N ′ /N ′
N′
f (x −1 γn′ x) dn′ χT γN ′ (x) dx,
where χT γN ′ (x) =
X
P γ∈Pinfl γ N =N ′
τPT (x).
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Let JCT,N ′ (f , λ) with λ ∈ a∗Q,C be the contribution from fixed N ′ T (x)−λ , where aT (qk) = exp(−T )qQ1 for with damping factor aQ Q Q q ∈ Q, k ∈ K. For example, the truncated zeta integral Z X T T aQ JC ,{1} (f , λ) = (l)−λ−2ρU/U ′ fU ′ (l −1 ξl) χT ξ (l) dl, L\L∩G1
ξ∈C ∩U/U ′
where U = exp u = C ∩ Qcan , U ′ = exp u′ = [U, U] and Z Z ′ f (k −1 xu ′ k) du ′ dk (x ∈ U/U ′ ). fU (x) = K
U′
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Let JCT,N ′ (f , λ) with λ ∈ a∗Q,C be the contribution from fixed N ′ T (x)−λ , where aT (qk) = exp(−T )qQ1 for with damping factor aQ Q Q q ∈ Q, k ∈ K. For example, the truncated zeta integral Z X T T aQ JC ,{1} (f , λ) = (l)−λ−2ρU/U ′ fU ′ (l −1 ξl) χT ξ (l) dl, L\L∩G1
ξ∈C ∩U/U ′
where U = exp u = C ∩ Qcan , U ′ = exp u′ = [U, U] and Z Z ′ f (k −1 xu ′ k) du ′ dk (x ∈ U/U ′ ). fU (x) = K
U′
Theorem (Wakatsuki-H.) For classical groups of absolute rank ≤ 2, the above transformation of JCT (f ) is valid, JCT,N ′ (f , λ) converges for Re λ ∈ (a∗Q )+ and can be continued meromorphically.
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Examples G = Sp(V , ω), dim V = 4, γ = exp X subregular unipotent: [2, 2] Canonical flag: V0 = Ker X = Im X .
V
Isomorphism X : V /V0 → V0 . Symmetric bilinear forms b+ on V /V0 , b− on V0 :
V0
b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ). 0
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Examples G = Sp(V , ω), dim V = 4, γ = exp X subregular unipotent: [2, 2] V
Canonical flag: V0 = Ker X = Im X . Isomorphism X : V /V0 → V0 .
U+ V0
Symmetric bilinear forms b+ on V /V0 , b− on V0 : b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
W+
U−
W− 0
Isotropic lines U+ /V0 , W+ /V0 for split b+ , U− , W− for split b− . ⊥ = U , XW = W ⊥ = W . XU+ = U+ − + − +
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Examples G = Sp(V , ω), dim V = 4, γ = exp X subregular unipotent: [2, 2] V
Canonical flag: V0 = Ker X = Im X . Isomorphism X : V /V0 → V0 .
U+ V0
Symmetric bilinear forms b+ on V /V0 , b− on V0 : b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
W+
U−
W− 0
Isotropic lines U+ /V0 , W+ /V0 for split b+ , U− , W− for split b− . ⊥ = U , XW = W ⊥ = W . XU+ = U+ − + − + ❅ Parabolics P with γ ∈ Pinfl : 7∅h ❅ −1 ′ (arrows N → N ) 0❅ 0 (U− , U+ ) (V0 ) (W− , W+ ) 1❅ ❅ P. V.: Quad(V0 ), AG /AQ ∼ = SO(V0 , b− ). γ
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G = GL(V ), dim V = 4, γ = exp X subregular unipotent: [3, 1] V
Canonical flag: V1 = Ker X ∩ Im X = Im X 2 , V2 = Ker X + Im X = Ker X 2 .
ww Ker XG G
V2 E E Im X V1 0
zz
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G = GL(V ), dim V = 4, γ = exp X subregular unipotent: [3, 1] V
Canonical flag: V1 = Ker X ∩ Im X = Im X 2 , V2 = Ker X + Im X = Ker X 2 . Parabolics P with γ ∈ Pinfl (arrows N → N ′ ):
(Im X )
jj4 ∅ jT?TT jjjj ???TTTTTTT j j j TT jjj
?? ?
(V1 )
?
?? ?
(V2 )
_?? ?
(Ker X )
(V1 , Im X ) (V1 , V2 ) (Ker X , V2 ) P.V.: Hom(V /V2 , V2 /V1 ) × Hom(V2 /V1 , V1 ) AG /AQ ∼ = GL(V1 ) γ
ww Ker XG G
V2 E E Im X V1 0
zz
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G = Sp(V , ω), dim V = 6, γ = exp X subregular unipotent: [4, 2] Canonical flag: 3
V 2
V+ = Ker X = Ker X + Im X ,
V+
V0 = Ker X 2 ∩ Im X = Ker X + Im X 2 = V0⊥ , V− = Ker X ∩ Im X 2 = Im X 3 = V+⊥ . Isom. X : V+ /V0 → V0 /V− , X 2 : V /V+ → V− . Symm. bil. forms b+ on V+ /V0 , b− on V0 /V− : b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
V0
V− 0
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G = Sp(V , ω), dim V = 6, γ = exp X subregular unipotent: [4, 2] Canonical flag: 3
V 2
V+ = Ker X = Ker X + Im X , V0 = Ker X 2 ∩ Im X = Ker X + Im X 2 = V0⊥ , V− = Ker X ∩ Im X 2 = Im X 3 = V+⊥ . X2
V+ Im X V0
Isom. X : V+ /V0 → V0 /V− , : V /V+ → V− . Symm. bil. forms b+ on V+ /V0 , b− on V0 /V− :
Ker X
b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
0
Nonisotropic lines Im X /V0 for b+ , Ker X /V− for b− ,
V−
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G = Sp(V , ω), dim V = 6, γ = exp X subregular unipotent: [4, 2] Canonical flag: 3
V 2
V+ = Ker X = Ker X + Im X , V0 = Ker X 2 ∩ Im X = Ker X + Im X 2 = V0⊥ , V− = Ker X ∩ Im X 2 = Im X 3 = V+⊥ . X2
V+ U+ Im X W+ V0
Isom. X : V+ /V0 → V0 /V− , : V /V+ → V− . Symm. bil. forms b+ on V+ /V0 , b− on V0 /V− :
U− Ker X W−
b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
0
Nonisotropic lines Im X /V0 for b+ , Ker X /V− for b− , Isotropic lines U+ /V0 , W+ /V0 for split b+ , U− /V− , W− /V− for split b− . ⊥ = U , XW = W ⊥ = W . XU+ = U+ − + − +
V−
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P.V.: Hom(V /V+ , V0 /V− ) × Quad(V0 /V− ) Parabolics P with γ ∈ Pinfl : ll5 ∅ lll ::: l l l :: lll :: lll l l :: ll l l l
(Ker X , Im X )
66 66 66 66 66
(V0 )
D
(Ker X , V0 , Im X ) N′
66 66 66 66 66
(V− , V+ )
(U− , U+ ) (W− , W+ )
(V− , U− , U+ , V+ )
(V− , V0 , V+ )
(V− , W− , W+ , V+ )
given by arrows except for (U− , U+ ): n′ = {Z ∈ n | ZV ⊂ U− , ZU+ = 0}, for (V− , U− , U+ , V+ ): n′ = {Z ∈ n | ZU+ ⊂ V− , ZV+ ⊂ U− }, for (W− , W+ ) and (V− , W− , W+ , V+ ) by analogy.
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The trace formula and prehomogeneous vector spaces Werner Hoffmann Bielefeld University, SFB 701
Simons Symposium on Families of Automorphic Forms and the Trace Fromula January 29, 2014
examples
induction
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can. parab.
mean values
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reordering
Induction of conjugacy classes Consider connected linear algebraic groups over a field F and geometric conjugacy classes. Theorem (Lusztig-Spaltenstein for unipotent classes, H.) Let P be a parabolic subgroup of a reductive group G with Levi decomposition P = MN, and C a conjugacy class in M. Then there is a unique dense P-conjugacy class C ′ in CN and a ˜ in G such that C ˜ ∩ P = C ′. unique conjugacy class C S P P ′ ˜ = IndG Notation C P C , C = InflM C , Pinfl = C InflM C
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Induction of conjugacy classes Consider connected linear algebraic groups over a field F and geometric conjugacy classes. Theorem (Lusztig-Spaltenstein for unipotent classes, H.) Let P be a parabolic subgroup of a reductive group G with Levi decomposition P = MN, and C a conjugacy class in M. Then there is a unique dense P-conjugacy class C ′ in CN and a ˜ in G such that C ˜ ∩ P = C ′. unique conjugacy class C S P P ′ ˜ = IndG Notation C P C , C = InflM C , Pinfl = C InflM C Theorem (Lusztig-Spaltenstein, H.)
G If M is a Levi component of both P and Q, then IndG P C = IndQ C . Given γ ∈ G , there are only finitely many P such that γ ∈ Pinfl .
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Prehomogeneous varieties Definition • A prehomogeneous G -variety is an irreducible variety V with an algebraic G -action such that there exists a dense G -orbit O. • It is called special if every relatively G -invariant rational function (defined over any extension of F ) is constant. • (V , G ) is called a prehomogeneous vector space if V is a vector space and the G -action is linear. • It is called regular if V ∗ is prehomogeneous and dp/p : O → V ∗ is a dominant morphism for some relative invariant p. p(gx) = χ(g )p(x).
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Prehomogeneous varieties Definition • A prehomogeneous G -variety is an irreducible variety V with an algebraic G -action such that there exists a dense G -orbit O. • It is called special if every relatively G -invariant rational function (defined over any extension of F ) is constant. • (V , G ) is called a prehomogeneous vector space if V is a vector space and the G -action is linear. • It is called regular if V ∗ is prehomogeneous and dp/p : O → V ∗ is a dominant morphism for some relative invariant p. p(gx) = χ(g )p(x). Zeta integral (F number field, A ring of adeles): Z X φ(g ξ) dg |χ1 (g )|s1 . . . |χr (g )|sr Z (φ, s1 , . . . , sr ) = G (A)/G (F )
ξ∈O(F )
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Canonical parabolic subgroups G reductive, char F = 0. Jacobson-Morozov: ∀ X ∈ g nilpotent, ∃ H, Y ∈ g such that [H, X ] = 2X ,
[H, Y ] = −2Y ,
[X , Y ] = H.
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Canonical parabolic subgroups G reductive, char F = 0. Jacobson-Morozov: ∀ X ∈ g nilpotent, ∃ H, Y ∈ g such that [H, Y ] = −2Y ,
[H, X ] = 2X ,
[X , Y ] = H.
Canonical parabolic of X and of exp X ∈ G (F ): M q= gn , Q = NormG q, n≥0
where gn = {Z ∈ g | [H, Z ] = nZ }.
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Canonical parabolic subgroups G reductive, char F = 0. Jacobson-Morozov: ∀ X ∈ g nilpotent, ∃ H, Y ∈ g such that [H, Y ] = −2Y ,
[H, X ] = 2X ,
[X , Y ] = H.
Canonical parabolic of X and of exp X ∈ G (F ): M q= gn , Q = NormG q, n≥0
where gn = {Z ∈ g | [H, Z ] = nZ }. Theorem (Vinberg, H.) L L Let u = n≥2 gn , u′ = [u, u] = n>2 gn . Then u/u′ is a regular prehomogeneous vector space under L = CentG H.
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Mean value formula Theorem (Siegel/Weil/Ono) If O is a special G -homogeneous variety over a number field F with trivial π1 (O(C)), π2 (O(C)) and X (G ), and if [G (A)ξ ∩ O(F ) : G (F )] is constant on O(F ), then Z Z X φ(g ξ) dg = φ(x) dx. G (A)/G (F ) ξ∈O(F )
G (A)O(F )
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Mean value formula Theorem (Siegel/Weil/Ono) If O is a special G -homogeneous variety over a number field F with trivial π1 (O(C)), π2 (O(C)) and X (G ), and if [G (A)ξ ∩ O(F ) : G (F )] is constant on O(F ), then Z Z X φ(g ξ) dg = φ(x) dx. G (A)/G (F ) ξ∈O(F )
G (A)O(F )
Conjecture There is a normal unipotent subgroup N C of P such that, for γ ∈ C ′ = InflP C , all elements of γN C ∩ C ′ have the same canonical parabolic and γN/N C is special under PγN .
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The geometric side of the trace formula Always write G for G (F ) and G for G (A). Exception: K a maximal compact subgroup of G such that G (F∞ )K is open and G = PK for every parabolic subgroup P. G1 the maximal closed normal subgroup with torsion-free abelian quotient.
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The geometric side of the trace formula Always write G for G (F ) and G for G (A). Exception: K a maximal compact subgroup of G such that G (F∞ )K is open and G = PK for every parabolic subgroup P. G1 the maximal closed normal subgroup with torsion-free abelian quotient. Representation RP of G1 in L2 (NP\G1 ): Z KP (x, y )φ(y ) dy RP (f )φ (x) = G \G1
for f ∈ Cc∞ (G1 ), φ ∈ L2 (NP\G1 ), where X Z f (x −1 γny ) dn. KP (x, y ) = γ∈P/N
N
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Truncation parameter TP ∈ aP = Lie(P/P1 ) Modular character ∆P : P → R+
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Truncation parameter TP ∈ aP = Lie(P/P1 ) Modular character ∆P : P → R+ For P maximal: Let −ˆ τPT be the characteristic function of {p ∈ P | ∆P (p) > ∆P (exp TP )}K For general P, τˆPT (x) =
Y
P ′ ⊃P P ′ max.
τˆPT′ (x)
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Truncation parameter TP ∈ aP = Lie(P/P1 ) Modular character ∆P : P → R+ For P maximal: Let −ˆ τPT be the characteristic function of {p ∈ P | ∆P (p) > ∆P (exp TP )}K For general P, τˆPT (x) =
Y
τˆPT′ (x)
P ′ ⊃P P ′ max.
Compatible family T = (TP )P : T ˆPT (x). τˆγPγ −1 (γx) = τ
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Arthur’s trace distribution: Z J T (f ) =
G \G1
X P
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KP (x, x)ˆ τPT (x) dx
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Arthur’s trace distribution: Z J T (f ) =
G \G1
X
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KP (x, x)ˆ τPT (x) dx
P
For geometric conjugacy class C in G , define JCT (f ) using X Z f (x −1 γny ) dn. KP,C (x, y ) = γ∈P/N IndG γ=C
N
Conjecture XZ C
hence
G \G1
X KP,C (x, x)ˆ τPT (x) dx < ∞,
J T (f ) =
P
P
C
JCT (f ).
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Reordering the unipotent contribution C a unipotent geometric conjugacy class. Z Z X X Z T JC (f ) = f (x −1 γnn′ x) dn′ dn τPT (x) dx. G \G1 P
γ∈P/N IndG γ=C
N/Nγ
Nγ
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Reordering the unipotent contribution C a unipotent geometric conjugacy class. Z Z X X Z T JC (f ) = f (x −1 γnn′ x) dn′ dn τPT (x) dx. G \G1 P
γ∈P/N IndG γ=C
N/Nγ
Nγ
R P For γ ∈ D with InflP D = D ′ replace γN/ND by (D ′ ∩γN)N D /N D using mean-value formula. Ignoring convergence: Z Z X X X T f (x −1 γn′ x) dn′ τPT (x) dx, JC (f ) = G \G1 P D ′ ⊂C ∩P γ∈D ′ N D /N D
where D = D ′ N/N.
ND
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Collect pairs (P, D ′ ) with given N ′ = N D , collect cosets γN D with given canonical parabolic Q (abbreviated γ ∈ Qcan ): Z XX X XZ T JC (f ) = f (x −1 γn′ x) dn′ τPT (x) dx. G \G1 Q N ′ ⊂Q γ∈(C ∩Q )N ′ /N ′ can
N′ P γ∈Pinfl N γ =N ′
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Collect pairs (P, D ′ ) with given N ′ = N D , collect cosets γN D with given canonical parabolic Q (abbreviated γ ∈ Qcan ): Z XX X XZ T JC (f ) = f (x −1 γn′ x) dn′ τPT (x) dx. G \G1 Q N ′ ⊂Q γ∈(C ∩Q )N ′ /N ′ can
N′ P γ∈Pinfl N γ =N ′
Fix a canonical parabolic Q for C : Z Z X X T JC (f ) = Q\G1
N ′ ⊂Q γ∈(C ∩Qcan )N ′ /N ′
N′
f (x −1 γn′ x) dn′ χT γN ′ (x) dx,
where χT γN ′ (x) =
X
P γ∈Pinfl γ N =N ′
τPT (x).
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Let JCT,N ′ (f , λ) with λ ∈ a∗Q,C be the contribution from fixed N ′ T (x)−λ , where aT (qk) = exp(−T )qQ1 for with damping factor aQ Q Q q ∈ Q, k ∈ K. For example, the truncated zeta integral Z X T T aQ JC ,{1} (f , λ) = (l)−λ−2ρU/U ′ fU ′ (l −1 ξl) χT ξ (l) dl, L\L∩G1
ξ∈C ∩U/U ′
where U = exp u = C ∩ Qcan , U ′ = exp u′ = [U, U] and Z Z ′ f (k −1 xu ′ k) du ′ dk (x ∈ U/U ′ ). fU (x) = K
U′
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Let JCT,N ′ (f , λ) with λ ∈ a∗Q,C be the contribution from fixed N ′ T (x)−λ , where aT (qk) = exp(−T )qQ1 for with damping factor aQ Q Q q ∈ Q, k ∈ K. For example, the truncated zeta integral Z X T T aQ JC ,{1} (f , λ) = (l)−λ−2ρU/U ′ fU ′ (l −1 ξl) χT ξ (l) dl, L\L∩G1
ξ∈C ∩U/U ′
where U = exp u = C ∩ Qcan , U ′ = exp u′ = [U, U] and Z Z ′ f (k −1 xu ′ k) du ′ dk (x ∈ U/U ′ ). fU (x) = K
U′
Theorem (Wakatsuki-H.) For classical groups of absolute rank ≤ 2, the above transformation of JCT (f ) is valid, JCT,N ′ (f , λ) converges for Re λ ∈ (a∗Q )+ and can be continued meromorphically.
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Examples G = Sp(V , ω), dim V = 4, γ = exp X subregular unipotent: [2, 2] Canonical flag: V0 = Ker X = Im X .
V
Isomorphism X : V /V0 → V0 . Symmetric bilinear forms b+ on V /V0 , b− on V0 :
V0
b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ). 0
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Examples G = Sp(V , ω), dim V = 4, γ = exp X subregular unipotent: [2, 2] V
Canonical flag: V0 = Ker X = Im X . Isomorphism X : V /V0 → V0 .
U+ V0
Symmetric bilinear forms b+ on V /V0 , b− on V0 : b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
W+
U−
W− 0
Isotropic lines U+ /V0 , W+ /V0 for split b+ , U− , W− for split b− . ⊥ = U , XW = W ⊥ = W . XU+ = U+ − + − +
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Examples G = Sp(V , ω), dim V = 4, γ = exp X subregular unipotent: [2, 2] V
Canonical flag: V0 = Ker X = Im X . Isomorphism X : V /V0 → V0 .
U+ V0
Symmetric bilinear forms b+ on V /V0 , b− on V0 : b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
W+
U−
W− 0
Isotropic lines U+ /V0 , W+ /V0 for split b+ , U− , W− for split b− . ⊥ = U , XW = W ⊥ = W . XU+ = U+ − + − + ❅ Parabolics P with γ ∈ Pinfl : 7∅h ❅ −1 ′ (arrows N → N ) 0❅ 0 (U− , U+ ) (V0 ) (W− , W+ ) 1❅ ❅ P. V.: Quad(V0 ), AG /AQ ∼ = SO(V0 , b− ). γ
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G = GL(V ), dim V = 4, γ = exp X subregular unipotent: [3, 1] V
Canonical flag: V1 = Ker X ∩ Im X = Im X 2 , V2 = Ker X + Im X = Ker X 2 .
ww Ker XG G
V2 E E Im X V1 0
zz
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G = GL(V ), dim V = 4, γ = exp X subregular unipotent: [3, 1] V
Canonical flag: V1 = Ker X ∩ Im X = Im X 2 , V2 = Ker X + Im X = Ker X 2 . Parabolics P with γ ∈ Pinfl (arrows N → N ′ ):
(Im X )
jj4 ∅ jT?TT jjjj ???TTTTTTT j j j TT jjj
?? ?
(V1 )
?
?? ?
(V2 )
_?? ?
(Ker X )
(V1 , Im X ) (V1 , V2 ) (Ker X , V2 ) P.V.: Hom(V /V2 , V2 /V1 ) × Hom(V2 /V1 , V1 ) AG /AQ ∼ = GL(V1 ) γ
ww Ker XG G
V2 E E Im X V1 0
zz
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G = Sp(V , ω), dim V = 6, γ = exp X subregular unipotent: [4, 2] Canonical flag: 3
V 2
V+ = Ker X = Ker X + Im X ,
V+
V0 = Ker X 2 ∩ Im X = Ker X + Im X 2 = V0⊥ , V− = Ker X ∩ Im X 2 = Im X 3 = V+⊥ . Isom. X : V+ /V0 → V0 /V− , X 2 : V /V+ → V− . Symm. bil. forms b+ on V+ /V0 , b− on V0 /V− : b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
V0
V− 0
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G = Sp(V , ω), dim V = 6, γ = exp X subregular unipotent: [4, 2] Canonical flag: 3
V 2
V+ = Ker X = Ker X + Im X , V0 = Ker X 2 ∩ Im X = Ker X + Im X 2 = V0⊥ , V− = Ker X ∩ Im X 2 = Im X 3 = V+⊥ . X2
V+ Im X V0
Isom. X : V+ /V0 → V0 /V− , : V /V+ → V− . Symm. bil. forms b+ on V+ /V0 , b− on V0 /V− :
Ker X
b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
0
Nonisotropic lines Im X /V0 for b+ , Ker X /V− for b− ,
V−
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G = Sp(V , ω), dim V = 6, γ = exp X subregular unipotent: [4, 2] Canonical flag: 3
V 2
V+ = Ker X = Ker X + Im X , V0 = Ker X 2 ∩ Im X = Ker X + Im X 2 = V0⊥ , V− = Ker X ∩ Im X 2 = Im X 3 = V+⊥ . X2
V+ U+ Im X W+ V0
Isom. X : V+ /V0 → V0 /V− , : V /V+ → V− . Symm. bil. forms b+ on V+ /V0 , b− on V0 /V− :
U− Ker X W−
b+ (u, v ) = ω(u, Xv ) = b− (Xu, Xv ).
0
Nonisotropic lines Im X /V0 for b+ , Ker X /V− for b− , Isotropic lines U+ /V0 , W+ /V0 for split b+ , U− /V− , W− /V− for split b− . ⊥ = U , XW = W ⊥ = W . XU+ = U+ − + − +
V−
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P.V.: Hom(V /V+ , V0 /V− ) × Quad(V0 /V− ) Parabolics P with γ ∈ Pinfl : ll5 ∅ lll ::: l l l :: lll :: lll l l :: ll l l l
(Ker X , Im X )
66 66 66 66 66
(V0 )
D
(Ker X , V0 , Im X ) N′
66 66 66 66 66
(V− , V+ )
(U− , U+ ) (W− , W+ )
(V− , U− , U+ , V+ )
(V− , V0 , V+ )
(V− , W− , W+ , V+ )
given by arrows except for (U− , U+ ): n′ = {Z ∈ n | ZV ⊂ U− , ZU+ = 0}, for (V− , U− , U+ , V+ ): n′ = {Z ∈ n | ZU+ ⊂ V− , ZV+ ⊂ U− }, for (W− , W+ ) and (V− , W− , W+ , V+ ) by analogy.
