Archimedean zeta integrals for GL(3,R) × GL (2,R)

Archimedean zeta integrals for GL(3, R) × GL(2, R)

Taku Ishii (Seikei University) Reduction to radial parts

• G≡Gn =GL(n,R),K≡Kn=O(n): maximal compact subgroup ofG

• N ≡Nn ={(xij)∈Gn|xii= 1, xij= 0 (i > j)}: maximal unipotent subgroup ofG

• G=N AK: Iwasawa decomposition with A≡An={y= diag(y1y2· · ·yn, y2· · ·yn, . . . , yn)|yi>0}

• ψ_R(t) = exp(2π√

−1t) (t∈R),ψ_N(x) =ψ_R(x₁₂+· · ·+x_n−1,n) (x= (x_ij)∈N)

• (π, Hπ): irreducible admissible representation ofG

• C^∞(N\G, ψN) ={f ∈C^∞(G,C)|f(xg) =ψN(x)f(g),∀(x, g)∈N×G}

• Iπ,ψ:={Φ|Φ∈Hom_(g,K)(Hπ,K, C^∞(N\G, ψN)), Φ(v) is of moderate growth (v∈Hπ,K)}

• W(π, ψR) ={Φ(v)|Φ∈ Iπ,ψ, v∈Hπ,K}: Whittaker model ofπ

• (π, H_π), (π^′, H_π′): irreducible admissible generic representations ofG_n+1,G_n

• (τ, Vτ), (τ^′, Vτ^′): irreducible representations ofKn withKn-embeddingsφ:Vτ ,→ W(π, ψR),φ^′ :Vτ^′ ,→ W(π^′, ψ_R⁻¹) Lemma 1. ForW ∈ W(π, ψ_R)andW^′∈ W(π^′, ψ⁻_R¹) we set

Z(s, W, W^′) :=

∫

Nn\Gn

W (g

1 )

W^′(g)|detg|^s−1/2dg.

Forv∈V_τ andv^′∈V_τ′, we have

Z(s, φ(v), φ^′(v^′)) = ⟨v, v^′⟩ dimVτ

dim∑Vτ

i=1

∫

A_n

φ(vi) (y

1 )

φ^′(v^′_i)(y)|dety|^s−1/2δ(y)⁻¹dy

if τ^′ =bτ (contragredient of τ), and Z(s, φ(v), φ^′(v^′)) = 0 if τ^′ ≇bτ. Here {v_i} is a basis of V_τ and {v^′_i} is its dual basis of Vτ^′ =Vτˆ andδ(y) =∏n−1

i=1 yⁱ⁽ⁿ_i ^−i)/2. Explicit formulas forGL(2,R)-Whittaker functions

• π=D_(ν,l): discrete series ofG₂ (ν∈C, l∈Z_>0)

• λ∈Z>0

• (τ_λ⁽²⁾, V_λ⁽²⁾=Cv_λ⊕Cv_−λ)∈Kc₂ withK₂-actionτ_λ⁽²⁾( _{cosθ sinθ}

−sinθcosθ

)v_q =e^√−1qθv_q,τ_λ⁽²⁾(⁻¹₁)v_q =v_−q (q∈ {±λ})

• Kc2={1,det} ∪ {τ_λ⁽²⁾ |λ∈Z>0}

• D_(ν,l)|K2 ∼=⊕_∞

i=0τ_l+1+2i⁽²⁾

Proposition 2. There exsits aK2-embeddingφ:V_l+1⁽²⁾→ W(D(ν,l), ψ^ε_R) (ε∈ {±1})whose radial part is given by φ(vε′(l+1))(diag(y1y2, y2)) =δε,ε^′·y^1/2₁ y₂^2ν· 1

2π√

−1

∫

s

ΓC

(

s+ν+ l 2

)

y₁^−sds, (ε^′ ∈ {±1}), where∫

s is a vertical line fromRe(s)−√

−1∞toRe(s) +√

−1∞with the sufficiently large real part to keep the poles of the integrand on its left.

Explicit formulas forGL(3,R)-Whittaker functions

• π= Ind_P2,1(D_(ν1,l)⊠χ_(ν2,δ)): generalized principal series ofG₃ (ν₁, ν₂∈C, l∈Z_>0, δ∈ {0,1})

• λ= (λ₁, λ₂),λ₁∈Z_≥0,λ₂∈ {0,1}

• P_λ={degreeλ₁ homogeneous polynomial inz₁, z₂, z₃}

• K₃acts onP_λ by (T_λ(k)p)(z₁, z₂, z₃) = (detk)^λ2p((z₁, z₂, z₃)·k),k∈K₃,p∈P_λ

• (τ_λ⁽³⁾, V_λ⁽³⁾)∈Kc3: quotient representation of Tλ onV_λ⁽³⁾=Pλ/(z²₁+z₂²+z²₃)P_λ−(2,0)

• Sλ={m= (m1, m2, m3)∈(Z_≥0)³|m1+m2+m3=λ1}

• {u^λ_m= image ofz₁^m1z^m₂²z^m₃³ under P_λ→V_λ⁽³⁾|m= (m₁, m₂, m₃)∈S_λ}: generator of V_λ⁽³⁾

• {v^λ_q := image of ((sgnq)z1+√

−1z2)^|q|z^λ₃^1−|q|| −λ1≤q≤λ1}: basis ofV_λ⁽³⁾

• τ_λ⁽³⁾

( _{cosθ sinθ}

−sinθcosθ 1

)

v^λ_q =e^√−1qθv^λ_q,τ_λ⁽³⁾(diag(ε₁, ε₂, ε₃))v_q^λ=ε^λ₁²ε^λ₂^2+qε^λ₃^1+λ2+qv^λ_ε

1ε₂q (ε_i∈ {±1})

• τ_λ⁽³⁾|K2 ∼= det^λ2⊕⊕λ1

i=1τ_i⁽²⁾, τ_i⁽²⁾∋v_±i→v^λ_±i∈τ_λ⁽³⁾: K₂-injection

• π|K3 ∼=τ_(l+1,δ)⁽³⁾ ⊕τ_(l+2,1⁽³⁾ _−δ)⊕⊕_∞

i=2m(i)τ(l+1+i,δ(i))⁽³⁾ withδ(i)≡δ+i+ 1 (mod 2) andm(i)≥2 (multiplicity) Theorem 3. [1]There exists aK3-embeddingφ:V_(l+1,δ)⁽³⁾ → W(π, ψR) whose radial part is given by

φ(u^λ_m)(diag(y1y2y3, y2y3, y3)) = (√

−1)^m1−m3y1y2(y2y3)^2ν1+ν2· 1 (2π√

−1)²

∫

s2

∫

s1

Γm(s1, s2)y⁻₁^s1y₂^−s2ds1ds2,

where

Γm(s1, s2) =ΓC(s1+ν1+₂^l)ΓR(s1+ν2+m1)ΓC(s2−ν1+₂^l)ΓR(s2−ν2+m3) ΓR(s1+s2+m1+m3) .

1

Computation of archimedean zeta integrals forG₃×G₂

• π= Ind_P2,1(D_(ν1,l)⊠χ_(ν2,δ)),π^′=D_(ν′,l^′) withl≥l^′

• Vτ=V_l⁽²⁾_′+1=Cvl^′+1⊕Cv_−(l′+1)

• φ:V_l⁽²⁾′+1,→V_(l+1,δ)⁽³⁾ →W(π, ψR)

• φ^′:V_l⁽²⁾_′+1→ W(π^′, ψ⁻_R¹)

Theorem 4. [2]Forv, v^′∈V_l⁽²⁾_′+1, we have Z(s, φ(v), φ(v^′)) =.⟨v, v^′⟩(√

−1)^2l′−l+1ΓC(s+ν1+ν^′+l+l^′

2 )ΓC(s+ν1+ν^′+l−l^′

2 )ΓC(s+ν2+ν^′+l^′ 2).

(outline of proof) We have

Z(s)≡Z(s, φ(v), φ^′(v^′)) = ⟨v, v^′⟩ 2

∫ _∞

0

∫ _∞

0

φ(vl^′+1)(diag(y1y2, y2,1))φ^′(v_−(l′+1))(diag(y1y2, y2))y₁^s−3/2y₂^2s−1dy1

y1

dy2

y2

Since (z1+√

−1z2)^l′+1z₃^l−l′ =∑l^′+1 j=0

(_l′+1 j

)√−1^l′+1−jz₁^jz₂^l′+1−jz₃^l−l′, Mellin-inversion implies that

Z(s) =⟨v, v^′⟩ 2

√−1^2l

′−l+1

ΓC(2s+ν1+ν2+ 2ν^′+ l

2)ΓR(2s+ 2ν1+ 2ν^′+l−l^′)

l^′+1

∑

j=0

(l^′+ 1 j

)

× 1 2π√

−1

∫

t

Γ_C(s−t+ν₁+₂^l)Γ_R(s−t+ν₂+j)

Γ_R(3s−t+ 2ν₁+ν₂+ 2ν^′+j+l−l^′)·Γ_C(t+ν^′+l^′ 2)dt.

Substitutingt→t+j and using the formula

∑m

j=0

(m j

)

ΓC(a+j)ΓC(b−j) =ΓC(a)ΓC(b−m)ΓC(a+b) ΓC(a+b−m) , we know that

Z(s) =⟨v, v^′⟩ 2

√−1^2l

′−l+1

Γ_C(2s+ν₁+ν₂+ 2ν^′+ l

2)Γ_R(2s+ 2ν₁+ 2ν^′+l−l^′)· Γ_C(s+ν₁+ν^′+^l+l₂^′) ΓC(s+ν1+ν^′+^l−₂^l′ −1)

× 1 2π√

−1

∫

t

Γ_R(s−t+ν₂)Γ_C(t+ν^′+^l₂^′)Γ_C(s−t+ν₁+₂^l −l^′−1) ΓR(3s−t+ 2ν1+ν2+ 2ν^′+l−l^′) dt.

From the formulas Γ_C(s) = Γ_R(s)Γ_R(s+ 1) (duplation formula) and 1

4π√

−1

∫

t

ΓR(t+a)ΓR(t+b)ΓR(t+c)ΓR(−t+d)ΓR(−t+e) Γ_R(t+a+b+c+d+e) dt

=Γ_R(a+d)Γ_R(b+d)Γ_R(c+d)Γ_R(a+e)Γ_R(b+e)Γ_R(c+e) ΓR(a+b+d+e)ΓR(a+c+d+e)ΓR(b+c+d+e) (Barnes’ second lemma), we can get the assertion. 2

• Barnes’ frist lemma:

1 4π√

−1

∫

t

Γ_R(t+a)Γ_R(t+b)Γ_R(−t+c)Γ_R(−t+d)dt=Γ_R(a+c)Γ_R(a+d)Γ_R(b+c)Γ_R(b+d) ΓR(a+b+c+d)

Remark 1. The casel^′> l:

• Eij ∈gl(3,R): matrix unit

• λ∈Z>0, Dλ:=C-span{Eλ= (E23−√

−1E13)^λ, E₋λ= (E23+√

−1E13)^λ} ⊂U(gl(3,C))

• Dλ∼=V_λ⁽²⁾ as K2-module (adjoint action onDλ) viaE_±λ↔v_±λ

• φ:V_l⁽²⁾_′+1,→V_l⁽²⁾_′−l⊗V_l+1⁽²⁾→W(π, ψ_R),φ^′ :V_l⁽²⁾_′+1→ W(π^′, ψ_R⁻¹) References

[1] Tadashi Miyazaki, Whittaker functions for generalized principal series representations of SL(3,R). Manuscripta Math., 128(2009), 107–135.

[2] Miki Hirano, Taku Ishii and Tadashi Miyazaki, The archimedean zeta integrals for GL(3)×GL(2). Proc. Japan Acad.,92. Ser. A (2016), 27–32.

Faculty of Science and Technology, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino, Tokyo, 180-8633, Japan E-mail address: ishii@st.seikei.ac.jp

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