Twisted Radon Transforms on Grassmannians

Toshio OSHIMA October 8, 2001

“Integral Geometry in Representation Theory” MSRI, Berkeley §1. Differential Equations Satisfied by Functions on Grassmannians

𝕏_{k}^{n} := {k-subspaces in 𝔽^{n}}, 𝔽 = ℝ, ℂ, ℍ

M^{o}(n, k; 𝔽) := \{X = (\begin{matrix}x_{11} & ⋯ & x_{1k} \\ ⋮ & ⋯ & ⋮ \\ x_{n1} & ⋯ & x_{nk} \end{matrix}) ∈ M(n, k; 𝔽); rank X = k\}

= M^{o}(n, k; 𝔽)/GL(k, 𝔽)

dim_{ℝ} 𝕏_{k}^{n} = (nk − k^{2} = (n − k)k) × \begin{cases}1 𝔽 = & ℝ, \\ 2 & ℂ, \\ 4 & ℍ. \end{cases}

ℙ^{n−1}(𝔽) := 𝕏_{1}^{n} = 𝔽^{n}/𝔽^{×} : A projective space

The group G = GL(n, 𝔽) acts on 𝕏_{k}^{n} (by left) and then

𝕏_{k}^{n} = GL(n, 𝔽)/P_{k,n}  (= O(n)/O(k) × O(n − k) if 𝔽 = ℝ)
P_{k,n} := \{p = (\begin{matrix}g_{1} & 0 \\ y & g_{2} \end{matrix}); g_{1} ∈ GL(k, 𝔽), g_{2} ∈ GL(n − k, 𝔽), y ∈ M(n − k, k; 𝔽)\}
B(G/P_{k,n};L_{λ}) := {f ∈ B(G); f(xp) = f(x)| det g_{1}|^{λ_{1}}| det g_{2}|^{λ_{2}}, ∀p ∈ P_{k,n}}
(= B(O(n)/O(k) × O(n − k)) if 𝔽 = ℝ)

= {f ∈ B(M^{o}(n, k; 𝔽)); f(Xg_{1}) = f(X)| det g_{1}|^{−λ_{1}}, ∀g_{1} ∈ GL(k, 𝔽)}

(x → ^{t}x^{−1} → X)

𝔤 : = ℂ ⊗ Lie(G)  (≃ M(n, ℂ) if G = GL(n; ℝ))
(Xf)(x) := \frac{d}{dt}f(xe^{tX})|_{t=0}  (E_{ij} = n ; ∑ ; ν=1 x_{νi}\frac{∂}{∂x_{νj}} , (x_{ij}) ∈ G)
(L_{X}f)(x) := \frac{d}{dt}f(e^{−tX}x)|_{t=0}
U^{ϵ}(𝔤) := ( ∞ ; ⨁ ; k=0 ⊗^{k}𝔤) /⟨X ⊗ Y − Y ⊗ X − ϵ[X, Y]⟩

: Homogenized universal enveloping algebra

U^{ϵ}(𝔤) ≃ U(𝔤) := U^{1}(𝔤) (X ↔ ϵX) if ϵ ≠ 0. S(𝔤) := U^{0}(𝔤)
λ : 𝔭 (:= ℂ ⊗ Lie(P)) → ℂ
J_{𝔭}^{ϵ}(λ) := _{X∈𝔭}^{∑} U^{ϵ}(𝔤)(X − λ(X))

J_{𝔭}(λ) := J_{𝔭}^{1}(λ) is the left ideal of U(𝔤) which kills B(G/P_{k,n}; L_{λ}) by identifying U(𝔤) with the ring of left invariant differential operators on G. If we identify U(𝔤) with the ring of right invariant differential operators, the ideal killing B(G/P_{k,n}; L_{λ}) is given by

I_{𝔭}^{ϵ}(λ) := _{g∈G}^{⋂} Ad(g)J_{𝔭}^{ϵ}(λ) (two sided ideal of U^{ϵ}(𝔤))

with ϵ = 1 through the anti-automorphism (X ↦ −X, XY ↦ (−Y)(−X)).

Namely

M_{𝔭}^{ϵ}(λ) := U^{ϵ}(𝔤)/J_{𝔭}^{ϵ}(λ) (a generalized Verma module of scalar type  )
I_{𝔭}^{ϵ}(λ) = Ann_{U^{ϵ}(𝔤)}(M_{𝔭}^{ϵ}(λ)) if ϵ ≠ 0.

Problem: Give an explicit construction of the generator system of I_{𝔭}^{ϵ}(λ)! Idea: Consider the case when ϵ = 0 and quantize it!. We will identify 𝔤 and 𝔤^{∗} by ⟨X, Y ⟩ = Trace XY. Then J_{𝔭}^{0}(λ) is the defining ideal of the set

A_{𝔭,λ} := \{X = (\begin{matrix}λ_{1}I_{k} & 0 \\ Y & λ_{2}I_{n−k} \end{matrix}); Y ∈ M(n − k, k; ℂ)\}

and our problem is to find the generator system of the defining ideal of the closure of the conjugacy class of a matrix in M(n; ℂ).

f ∈ Ad(g)J_{𝔭}^{0}(λ), ∀g ∈ G ⇔ f(Ad(g)A_{𝔭,λ}) = 0, ∀g ∈ G.

Method I (minors, elementary devisors):

rank (X − λ_{1}) ≤ n − k ⇒ det (X − λ_{1})_{IJ} ∈ I_{𝔭}^{0}(λ)
with I = {i_{1}, …, i_{n−k+1} }, J = {j_{1}, …, j_{n−k+1} }.
I_{𝔭}^{0}(λ) = ⟨det (X − λ_{1})_{IJ}, det (X − λ_{2})_{I′J′}; #I = #J = n − k + 1,
#I′ = #J′ = k + 1⟩ if λ_{1} ≠ λ_{2}.

⇒ elementary divisors + quantization!

Theorem 1. Suppose 2k ≤ n. If λ_{1} − λ_{2} ∉ {kϵ, ⋯, (n − k)ϵ}, then

I_{𝔭}^{ϵ}(λ) = ⟨ det (E_{i_{μ}j_{ν}} − (λ_{1} + (ν − n + k − 1)ϵ)δ_{i_{μ}j_{ν}})\begin{matrix}1≤μ≤n−k+1 \\ 1≤ν≤n−k+1 \end{matrix} ,

det (E_{i′_{μ}j′_{ν}} − (λ_{2} + (ν − 1)ϵ)δ_{i′_{μ}j′_{ν}})\begin{matrix}1≤μ≤k+1 \\ 1≤ν≤k+1 \end{matrix} ⟩.

If λ_{1} − λ_{2} ∈ {kϵ, ⋯, (n − k)ϵ}, then

I_{𝔭}^{ϵ}(λ) = ⟨\frac{d}{dt} det (E_{i_{μ}j_{ν}} − (t + λ_{1} + (ν − n + k − 1)ϵ)δ_{i_{μ}j_{ν}})\begin{matrix}1≤μ≤n−k+1 \\ 1≤ν≤n−k+1 \end{matrix} |_{t=0} ,

det (E_{i′_{μ}j′_{ν}} − (λ_{2} + (ν − 1)ϵ)δ_{i′_{μ}j′_{ν}})\begin{matrix}1≤μ≤k+1 \\ 1≤ν≤k+1 \end{matrix} ⟩.

Here det (A_{ij}) = ∑_{σ} sign (σ)A_{σ(1)1} A_{σ(2)2} ⋯ and I = {i_{1}, …, i_{n−k−1}} etc. A similar result also holds in the case when 2k ≥ n.

Remark 1. i) The action of G on the space of the ℂ-span of the above (Capelli type) generators does not depend on ϵ. Hence the ℂ-span is a G-module. ii) This theorem is generalized for any parabolic 𝔭 of 𝔤𝔩_{n} and any λ (preprint, 2000). iii) ϵ = λ = 0 (⇒ nilpotent variety) conjectured by Tanisaki, proved by Weyman(1989) (by Kostant with moreover 𝔭 is a Borel subalgebra). iv) In the case when 𝔤 = 𝔬_{n}, a similar constuction is done by H. Oda (together with quantized Pfaffians).

Method II (minimal polynomials) The minimal polynomial of X ∈ A_{𝔭,λ} is q(x) = (x − λ_{1})(x − λ_{2}) and if ϵ = 0 and λ_{1} ≠ λ_{2}, we have

I_{𝔭}^{0}(λ) = ⟨q(X)_{ij}, Trace X − kλ_{1} − (n − k)λ_{2}; 1 ≤ i ≤ n, 1 ≤ j ≤ n⟩.

Theorem 2. Putting q^{ϵ}(x) = (x − λ_{1})(x − λ_{2} − kϵ), we have

I_{𝔭}^{ϵ}(λ) = ⟨q^{ϵ}(F)_{ij}, Trace F − kλ_{1} − (n − k)λ_{2}; 1 ≤ i ≤ n, 1 ≤ j ≤ n⟩
if \begin{cases}M_{𝔭}^{ϵ}(λ) has a regular infinitesimal character & (ϵ ≠ 0), \\ λ_{1} ≠ λ_{2} & (ϵ = 0). \end{cases}

Here F = (E_{ij}) ∈ M(n; 𝔤) ⊂ M(n; U^{ϵ}(𝔤)).

Remark 2. This theorem is generalized in the case of any generalized Verma module M_{𝔭}^{ϵ}(λ) of scalar type for any reductive Lie algebra 𝔤 (with some condition for λ). The minimal polynomials are explicitly given by [Oda-O, in preparation] attached to any finite dimensional faithful completely reducible representation of 𝔤. In the case of the classical Lie algebra 𝔤 defined by 𝔤 = 𝔤𝔩_{N}^{σ} with an involution σ, we may naturally choose F = (E_{ij} + σ(E_{ij})) ∈ M(N; 𝔤) (natural representation).

Theorem 3. Let 𝔟 be a Borel subalgebra of 𝔤 with 𝔟 ⊂ 𝔭. Put J^{ϵ}(λ) := ∑_{X∈𝔟} U^{ϵ}(𝔤)(X − λ(X)). Then

J_{𝔭}^{ϵ}(λ) = J^{ϵ}(λ) + I_{𝔭}^{ϵ}(λ) (GAP)

if λ is regular with ϵ ≠ 0 (or the centralizer of λ is contained in 𝔭 with ϵ = 0).

Remark 3. i) M(λ) := U(𝔤)/J(λ) is the usual Verma module and M_{𝔭}(λ) is its quotient. ii) The necessary and sufficient condition for (GAP) is obtained for any 𝔭 of 𝔤𝔩_{n}. iii) Theorem 3 holds for the two-sided ideal constructed by the minimal polynomial for the natural representation if 𝔤 is classical (with some more (possible) exceptional values for λ if 𝔤 is exceptional).

§2. Applications to Integral Transformations

Penrose Transformations

G_{ℂ}: a complex reductive Lie group with a real from G P_{ℂ}: a parabolic subgroup of G_{ℂ} V: a G-orbit in G_{ℂ}/P_{ℂ} O_{λ}: a holomorphic line bundle over G_{ℂ}/P_{ℂ} The image of any G-map: H_{V}^{∗} (O_{λ}) → {functions } satisfies the system of differential equations which we constructed because the ideal is two-sided.

Poisson transformations

G: a connected real semisimple Lie group with finite center K: a maximal compact subgroup of G P: a parabolic subgroup containing a minimal parabolic subgroup P_{min}

P_{λ} : B(G/P; L_{λ}) → (⊂ B(G/P_{min}; L_{λ}) ⟶^{P_{λ}^{min}}) A(G/K; M_{λ}).
f ↦  (P_{λ}f)(g) = ∫_{K} f(gk)dk

Here M_{λ} is a maximal ideal of the ring of invariant differential operators on G/K and A(G/K; M_{λ}) is its solution space.

Then if P_{λ}^{min} is bijective (⇔ e(λ) ≠ 0: the condition due to Helgason) and (GAP) is valid, the image of P_{λ} is characterized by our system because (GAP) assures that the function in B(G/P_{min}; L_{λ}) satisfying our system is in B(G/P; L_{λ}). In paticular, these assumptions are satisfiled for any G and P if λ = 0.

Remark 4. i) If G is classical and P is maximal, then our system defined from the minimal polynomial is of order two or three. ii) The images of D′, C^{∞} and L^{p}([Ben Saïd-Shimeno-O, in preparation]) are also characterized.

§3. Radon Transformations on Grassmanians Integrate the functions on ℓ-subspace in 𝔽^{n} over k-subspaces (k > ℓ):

R_{k}^{ℓ} : B(𝕏_{ℓ}^{n}) ∋ ϕ ↦ (R_{k}^{ℓ}ϕ)(g) = ∫_{O(k)/O(ℓ)×O(k−ℓ)} ϕ(gk)dk ∈ B(𝕏_{k}^{n}).

This is studied by Funk(1916), Gelfand, Helgason, Grinberg, Gonzalez, Kakehi etc.(cf. Helgason’s Book). For simplicity we assume 𝔽 = ℝ.

Fact: R_{k}^{ℓ} is lifted up a G-map: R_{k}^{ℓ} : B(G/P_{ℓ,n}; L_{(k,0)} ) → B(G/P_{k,n}; L_{(ℓ,0)} )

Theorem 4 ([O, 1996]). Suppose 0 < ℓ < k < n and ℓ + k < n. Then R_{k}^{ℓ} is a toplogical G-isomorphism onto the solution space of the system:

\{Φ((x_{ij})\begin{matrix}1≤i≤n \\ 1≤j≤k \end{matrix}) ∈ B(M^{0}(n, k; ℝ));
Φ(xg ) = | det g|^{−ℓ}Φ(x) for g ∈ GL(k, ℝ),
det (\frac{∂}{∂x_{i_{μ}j_{ν}}} )\begin{matrix}1≤μ≤ℓ+1 \\ 1≤ν≤ℓ+1 \end{matrix} Φ(x) = 0 (Capelli type )

for 1 ≤ i_{1} < ⋯ < i_{ℓ+1} ≤ n, 1 ≤ j_{1} < ⋯ < j_{ℓ+1} ≤ k}.

Remark 5. i) B can be replaced by D′, C^{∞} etc. ii) The above det () can be replaced by a single higer order equation (by Kakehi).

Lemma (Generalized Capelli identity). If [\frac{∂}{∂x_{ij}}, x_{st}] = ϵδ_{is}δ_{jt}, then

det ( n ; ∑ ; ν=1 x_{νi_{k}} \frac{∂}{∂x_{νj_{ℓ}}} + ϵ(m − ℓ)δ_{i_{k}j_{ℓ}})\begin{matrix}1≤k≤m \\ 1≤ℓ≤m \end{matrix}

= _{1≤ν_{1}<···<ν_{m}≤n}^{∑} det (x_{ν_{p}i_{q}})\begin{matrix}1≤p≤m \\ 1≤q≤m \end{matrix} · det (\frac{∂}{∂x_{ν_{p}i_{q}}} )\begin{matrix}1≤p≤m \\ 1≤q≤m \end{matrix}

for I = {i_{1}, …, i_{m}} and J = {j_{1}, …, j_{m}}. Remark 6. If m > n, the above equals 0. (m = n ⇒ Capelli identity).

§4. Hypergeometric Functions P: a parabolic subgroup of a real reductive Lie group G Q_{j}: closed subgroups of G so that Q_{j}\G/P have open cosets (j = 1, 2) λ, μ_{j}: characters of P, Q_{j}, respectively. ϕ_{1}: a function on G with ϕ_{1}(q_{1}xp) = μ_{1}(q_{1})λ(p)ϕ_{1}(x) for q_{1} ∈ Q_{1} and p ∈ P ϕ_{2}: a function on G with ϕ_{2}(q_{2}xp) = μ_{1}(q_{2})λ^{∗}(p)ϕ_{2}(x) for q_{2} ∈ Q_{2} and p ∈ P (λ^{∗} = −λ − 2ρ_{P})

Definition (Hypergeometric functions).

Φ_{ϕ_{1},ϕ_{2}}(x) := ∫_{K} ϕ_{1}(xk)ϕ_{2}(k)dk (= ∫_{K} ϕ(k)ϕ_{2}(x^{−1}k)dk)

Remark 7. i) Equations satsified by Φ(x): left action of Q_{1}, right action of Q_{2} and our system for B(G/P; L_{λ}). ⇒ the total system of differential equations ii) Q_{1} = Q_{2} = K ⇒ (zonal) spherical functions (ex. Lauricella’s F_{D} etc.)

P = P_{ℓ,n}, Q_{2} = P_{k,n}, λ = (k, 0), μ_{2} = (−ℓ, 0), ϕ_{2} is the kernel function of R_{k}^{ℓ}. Theorem 5. Let H be a connected subgroup of GL(n, ℝ) such that (H_{ℂ} ×

GL(ℓ, ℂ), ℂ^{n} ⊗ ℂ^{ℓ}) is a prehomogeneous vector space (⇔ ∃ an open orbit). Then the soutions of our total system are our hypergeometric functions (i.e. integral transforms) of the relative invariants on M^{o}(n, ℓ; ℝ). Example (Gelfand’s Hypergeometric functions). ℓ = 1. H = GL(1, ℝ)_{+} × ⋯ × GL(1, ℝ)_{+} and μ_{1} = (α_{1}, ⋯, α_{n}) with ∑_{j=1}^{n} α_{j} = −k.

Φ(α, x) = ∫_{t_{1}^{2}+···+t_{k}^{2}=1} n ; ∏ ; j=1 | k ; ∑ ; ν=1 t_{ν}x_{jν}|_{±}^{α_{j}} ω (Hypergeometric functions ).

Our total system is (Φ(x_{1}, ⋯, x_{k}) is an even function for column vectors x_{ν})

k ; ∑ ; j=1 x_{ij} \frac{∂Φ}{∂x_{ij}} = α_{j}Φ for 1 ≤ i ≤ n (H-action )

n ; ∑ ; ν=1 x_{νi}\frac{∂Φ}{∂x_{νj}} = −ℓδ_{ij}Φ for 1 ≤ i, j ≤ k (GL(k, ℝ)-action )
\frac{∂^{2}Φ}{∂x_{i_{1}j_{1}}x_{i_{2}j_{2}}} = \frac{∂^{2}Φ}{∂x_{i_{2}j_{1}}x_{i_{1}j_{2}}} for 1 ≤ i_{1} < i_{2} ≤ n, 1 ≤ j_{1} < j_{2} ≤ k (Capelli type ).

Remark 8. i) n = 4 and k = 2 in Example ⇒ Gauss Hypergeometric functions. ii) If the prehomogeneous vector space has finite orbits in M^{o}(n, ℓ; ℂ), then the total system is holonomic ([Tanisaki]).

§5. Twisted Radon Transformations Consider

R_{G/H}^{G/K} : F(G/K) → F(G/H), f ↦ (R_{G/H}^{G/K}f)(x) = ∫_{H/K∩H} f(xh)dh

with a certain function space F.

What happens if H_{g} := gHg^{−1} (is twisted) with g ∈ G ?

Example. G = SO(3) and K = H = SO(2).

If θ corresponds to the zero of a zonal spherical function, the twisted Radon transform has a kernel corresponding to the parameter of the zonal spherical function.

Other θ ⇒ Small divisor, Diophantine approximation…

What is the good value of θ ?

H_{g}/K ∩ H_{g} is totally geodesic submanifold of G/K ⇒ ?

θ H_{g} K

𝕏_{2}^{n} = G/K, G = U(n, 𝔽), K = U(2, 𝔽) × U(n − 2, 𝔽) 𝕏_{k}^{n} = G/H, G = U(n, 𝔽), H = (U(k, 𝔽) × U(n − k, 𝔽))′

I_{2,n−2} := (\begin{matrix}I_{2} \\ & −I_{n−2} \end{matrix}), I_{1,1,k−1,n−k−1} := (\begin{matrix}I_{1} \\ & −I_{1} \\ & & I_{k−1} \\ & & & −I_{n−k−1} \end{matrix} ),

K = {g ∈ G; g = I_{2,n−2} gI_{2,n−2} } H = {g ∈ G; g = I_{1,1,k−1,n−k−1} gI_{1,1,k−1,n−k−1} }

⇒ H/H ∩ K = ℙ^{k−1}(𝔽) × ℙ^{n−k−1}(𝔽) ⊂ 𝕏_{2}^{n} (totally geodesic).

Theorem 6 [Kurita-O, in preparation]. Suppose 2 ≤ k ≤ n − 2. The above twisted Radon transformation

R : B(G/K) ∋ f ↦ (Rf)(x) = ∫_{H/K∩H} f(xh)dh ∈ B(G/H)

is a topological isomorphism onto the image of the non-twisted Radon transformation if n ≠ 2k. If n = 2k, the odd functions are in the kernel and the same result holds on the even functions.

Remark 9. n = 4 and k = 2 and 𝔽 = ℂ ⇒ the simplest non-trivial case, H/K ∩ H = ℙ^{1}(ℂ) × ℙ^{1}(ℂ) and R : B(𝕏_{2}^{4}/ℤ_{2}) ≃ B(𝕏_{2}^{4}/ℤ_{2}).

The End

This presentation program is written by Toshio OSHIMA ftp: CTAN/dviware/dviout/