LIMITS OF BESSEL FUNCTIONS FOR ROOT SYSTEMS AS THE RANK TENDS TO INFINITY

. We study the asymptotic behaviour of Bessel functions associated of root systems of type A n − 1 and type B n with positive multiplicities as the rank n tends to inﬁnity. In both cases, we characterize the possible limit functions and the Vershik-Kerov type sequences of spectral parameters for which such limits exist. In the type A case, this generalizes known results about the approximation of the (positive-deﬁnite) Olshanski spherical functions of the space of inﬁnite-dimensional Hermitian matrices over F = R , C , H (with the action of the associated inﬁnite unitary group) by spherical functions of ﬁnite-dimensional spaces of Hermitian matrices. In the type B case, our re-sults include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to in-ﬁnity, and a classiﬁcation of the Olshanski spherical functions of the associated inductive limits.


Introduction
The asymptotic analysis of multivariate special functions has a long tradition in infinite dimensional harmonic analysis, tracing back to the work of Olshanski, Vershik, and Kerov, see [Ol90,OV96,VK82].Of particular interest in this context are the behaviour of spherical representations and the limits of spherical functions of increasing families of Gelfand pairs as specific dimensions tend to infinity.
Bessel functions associated with root systems generalize the spherical functions of Riemannian symmetric spaces of Euclidean type, which occur for special values of the multiplicity parameters.They appear naturally in rational Dunkl theory, with an intimate connection to the Dunkl kernel and the associated harmonic analysis.We refer to [Op93] for a general treatment of such Bessel functions and to [Ro03,dJ06,RV08,DX14] for an overview of rational Dunkl theory including the connection with symmetric spaces.There are two classes of particular interest, including applications to β-ensembles in random matrix theory, namely those of type A n−1 and type B n .We refer to [Fo10] for a general background and to [BCG22] for some recent developments.In the cases of type A and B, the Bessel functions can be expressed as hypergeometric series involving Jack polynomials, c.f. Section 2. Bessel functions of type A n−1 have a continuous multiplicity parameter k ≥ 0 and include as special cases the spherical functions of the motion groups U n (F) ⋉ H n (F) over F = R, C or H, where the unitary group U n (F) acts by conjugation on the space H n (F) of Hermitian matrices over F. These cases correspond to k = d 2 with d = dim R F ∈ {1, 2, 4}.Bessel functions of type B q have non-negative multiplicity parameters of the form κ = (k ′ , k), with k the multiplicity on the roots ±(e i ± e j ) and k ′ that on the roots ±e i .They generalize the spherical functions of the motion groups (U p (F) × U q (F)) ⋉ M p,q (F), with p ≥ q.Here the multiplicities are k = d 2 , k ′ = d 2 (p − q + 1) − 1 2 .In [RV13], the limits of the spherical functions of these motion groups as p → ∞ and the associated Olshanski spherical pairs were studied, where the rank q remained fixed.
In the present paper, we shall study Bessel functions of type A n−1 and type B n with arbitrary positive multiplicities as the rank tends to infinity, in the spirit of the work of Okounkov and Olshanski [OO98,OO06] about Jack polynomials (type A) and multivariate Jacobi polynomials (type BC).See also [Cu18] for a more recent extension of their results.
We obtain explicit asymptotic results for Bessel functions of type A and type B with arbitrary positive multiplicities as the rank goes to infinity.In the type A case, our results generalize those of [OV96] and [Bo07] for the limits of the spherical functions of the Gelfand pairs (U n (F) ⋉ H n (F), U n (F)) as n → ∞.In contrast to [Bo07], whose results in the geometric cases are also weaker than ours (c.f.Remark 3.6), we follow the direct approach of [OV96] for F = C via spherical expansions of the involved Bessel functions, which are replaced by hypergeometric expansions in terms of Jack polynomials in our general setting.To become more precise, we consider the Bessel functions J An−1 (iλ(n), (x, 0, . . ., 0)) with fixed multiplicity k > 0 and x ∈ R r for sequences of spectral parameters λ(n) ∈ R n as n → ∞.Following [OV96,OO98], we characterize those sequences (λ(n)) n∈N for which the associated sequence of Bessel functions converges (locally uniformly), in terms of specific real parameters α = (α i ) i∈N , β, γ with γ ≥ 0. These parameters describe the growth of the so-called Vershik-Kerov sequence (λ(n)) as n → ∞.In Theorem 3.5, the main result of Section 3, we obtain that for where the convergence is locally uniform on R r for each r ∈ N. In the group cases k = d 2 , the limiting functions are products of Polya functions (in the sense of [Fa08]).They coincide with the positive definite Olshanski spherical functions of the spherical pairs (U ∞ (F) ⋉ Herm ∞ (F), U ∞ (F)) which were already determined by Pickrell [Pi91]; see also [OV96], where they occur in the characterization of the ergodic measures of Herm ∞ (F) with respect to the action of U ∞ (F).
In the type B case, we consider Bessel functions J Bn (κ n , iλ(n), (x, 0, . . ., 0)) for n → ∞, where the multiplicity is of the form κ n = (k ′ n , k), i.e. the first multiplicity parameter may also vary with n.This is motivated by the geometric cases.Again we characterize the sequences (λ(n)) for which the associated Bessel functions converge locally uniformly on R r as n → ∞, and we determine the possible limits, which are now given by the functions with real parameters β ≥ 0 and α l ≥ 0 with n l=1 α l ≤ β.It turns out that for k = d 2 with d = 1, 2, 4, these can be identified with the positive definite Olshanski spherical functions of spherical pairs (G ∞ , K ∞ ), which are obtained as inductive limits of the motion groups (U p (F)×U q (F))⋉M p,q (F) as both dimension parameters p and q tend to infinity.
The organization of this paper is as follows: After a brief overview of Bessel functions associated with root systems in Section 2, the type A case is treated in Section 3.While in this case our results generalize known results in the geometric cases, our results for type B, which are developed in Section 4, seem to be new even in the geometric cases.

Bessel functions of type A and type B
In this section, we recall some basic facts about Bessel functions in Dunkl theory.We shall not go into details, but refer the reader to [DX14,Ro03] for some general background on Dunkl theory, and to [Op93,dJ06,RV08] for Dunkl-type Bessel functions and their relevance in the symmetric space context.For a reduced (but not necessarily essential) root system R ⊂ R n we fix a multiplicity function k : R → [0, ∞), i.e. k is invariant under the associated Weyl group W. Dunkl kernel, where E(λ, .) is characterized as the unique analytic solution of the joint eigenvalue problem for the associated rational Dunkl operators with spectral variable λ ∈ C n , normalized according to E(λ, 0) = 1.The kernel E is holomorphic on C n × C n and symmetric in its arguments.For each λ ∈ R n , there exists a compactly supported probability measure µ λ on R n such that (2.1) In particular, the kernel E(iλ, .) is positive-definite on the additive group R n .If k = 0, then E(iλ, x) = e i λ,x .The Bessel function associated with R and k is defined by It is W -invariant in both arguments and for λ ∈ R n , the function J(iλ, .) is also positive-definite on R n .We shall be concerned with the root systems where (e i ) denotes the standard basis of R n .In both cases, the Bessel functions can be written as hypergeometric series in terms of Jack polynomials.For A n−1 , the multiplicity function is given by a single parameter k ∈ [0, ∞).We write Λ + n for the set of partitions κ = (κ 1 , κ 2 , . . . ) of length l(κ) ≤ n and denote by C The Jack polynomials are stable with respect to the number of variables, i.e. for κ ∈ Λ + r with r < n we have uniquely extend to continuous functions C κ on C (∞) = ∞ n=1 C n , equipped with the inductive limit topology.We shall often consider elements from C (∞) as sequences x = (x n ) n∈N in C with x n = 0 for at most finitely many n; for R (∞) accordingly.
By [BF98], the Bessel function of type A n−1 with multiplicity k is given by the Jack hypergeometric series with the Jack polynomials of index α = 1/k as above.
Bessel functions of type A n−1 occur as the (zonal) spherical functions of the Gelfand pair (G n , K n ) := (U n (F) ⋉ Herm n (F), U n (F)), where the unitary group U n (F) over F = R, C, H acts by conjugation on the space Herm n (F) of Hermitian matrices over F. Recall that the spherical functions of a Gelfand pair (G, K) can be characterized as the continuous, K-biinvariant, non-zero functions ϕ on G satisfying the product formula The following characterization is possibly folklore, but not well documented in the literature.For the reader's convenience, we therefore provide a proof.
Lemma 2.1.The spherical functions of (G n , K n ), considered as U n (F)-invariant functions on Herm n (F), are given by the Bessel functions ϕ λ (X) = J An−1 d 2 ; iλ, σ(X) , λ ∈ C n where d = dim R F and σ(X) ∈ R n denotes the eigenvalues of X ∈ Herm n (F), decreasingly ordered by size and counted according to their multiplicity.Moreover, ϕ λ = ϕ µ iff there exists some w ∈ S n with µ = w.λ, and ϕ λ is positive definite iff λ ∈ R q .
Proof.Consider the Gelfand pair ( G n , K n ) = (SU n (F)⋉SHerm n (F), SU n (F)).Note that G n is the Cartan motion group of SL n (F), which is connected and semisimple.Thus by [dJ06, Sect.6] (c.f. also [RV08, Sect.3]), the spherical functions of ( G n , K n ) are given, as functions on SHerm n (F), by , where the inner product ., . is extended to C n × C n in a bilinear way.The restriction of π to R n is the orthogonal projection onto R n 0 = R n ∩ C n 0 .Then for z, w ∈ C n and arbitrary multiplicity k ≥ 0, J An−1 (k; z, w) = e z,1 n w,1 n /n • J An−1 (k; π(z), π(w)). (2.4) This follows e.g. from [BF98,Propos. 3.19].Now suppose ψ is a spherical function of (G n , K n ), considered as an S n -invariant function on R n .Then where x, y ∈ R n are identified with the corresponding n × n-diagonal matrices.It follows that where α ∈ C is a constant and the restriction of ψ to R n 0 is spherical for ( G n , K n ).Conversely, it is easily checked that each spherical function of ( G n , K n ) extends to a spherical function of (G n , K n ) in this way.The assertion now follows from (2.4).The assertion concerning the positive-definite spherical functions follows from [Wo06, Theorem 5.4].
For the root system R = B n , we denote the multiplicity (by slight abuse of notation) as (k ′ , k), where k is the value on the roots ±(e i ± e j ) and k ′ is the value Then the Bessel function of type B n with multiplicity (k ′ , k) can be written as with the hypergeometric series Here the squares in the arguments are understood componentwise and again, the Jack polynomials are those of index α = 1/k.It is easily seen that both 0 F 0 and 0 F 1 converge locally uniformly on C n × C n ; c.f. [BR23] for precise convergence properties of Jack hypergeometric series.Bessel functions of type B occur as the spherical functions of the Gelfand pairs (G, K) with where M p,q (F) is the space of p × q matrices over F = R, C, H and K acts on M p,q (F) via (U, V ).X = U XV −1 .The group G is the Cartan motion group of the non-compact Grassmann manifold U (p, q; F)/U p (F) × U q (F) which is of rank q.The spherical functions of (G, K) may be considered as K-invariant functions on M p,q (F) and thus depend only on the singular values of their argument.Again as a consequence of [dJ06], they are given by the Bessel functions F), ordered by size.We may therefore also consider the ϕ λ as functions on the closed Weyl chamber i.e. ϕ λ (x) = J Bq (κ; iλ, x), x ∈ C q .Moreover, ϕ λ = ϕ µ iff there exists some w ∈ W = S n ⋉ Z n 2 with µ = w.λ.The positive-definite spherical functions are the ϕ λ with λ ∈ C n , which again follows from [Wo06, Theorem 5.4].

The type A case
We start with some motivation from asymptotic spherical harmonic analysis, see [Ol90,Fa08] for a general background.Suppose that (G n , K n ), n ∈ N is an increasing sequence of Gelfand pairs, where where d n k is the normalized Haar measure on K n .We remark that this definition is according to [Fa08], whereas in [Ol90] spherical functions are in addition required to be positive definite.Consider now the sequence of Gelfand pairs (G n , K n ) = (U n (F) ⋉ Herm n (F), U n (F)) as above.We regard G n and K n as closed subgroups of G n+1 and K n+1 in the usual way.Then (G ∞ , K ∞ ) with the inductive limits is an Olshanski spherical pair.The positive definite spherical functions of (G ∞ , K ∞ ) were completely determined by Pickrell [Pi91, Sect.5], see also [OV96] for F = C, and [Fa08, Section 3].As functions on Herm ∞ (F), they are given by and (x 1 , x 2 , . . . ) ∈ R (∞) are the eigenvalues of X, ordered by size and counted according to their multiplicity.The product is invariant under rearrangements of the α l .For F = C it is also noted in [OV96] that the set of positive definite spherical functions is bijectively parametrized by the set In [OV96], explicit approximations of the positive definite spherical functions by positive definite spherical functions of the pairs (G n , K n ) with n → ∞ by use of spherical expansions were obtained in the case F = C.In [Bo07], this was generalized by completely different methods to F = R, H.
In the present paper, we shall obtain the result of Pickrell and explicit approximations of Olshanski spherical functions as particular cases of a more general asymptotic result for Bessel functions of type A n−1 with an arbitrary multiplicity parameter k > 0.
Let us first turn to the spectral parameters to be considered for n → ∞.Instead of working with multisets, it will be convenient for us to work with sequences (or finite tuples) with a prescribed order of their components.We introduce the following order on R: x ≪ y iff either |x| < |y| or |x| = |y| and x ≤ y.
Proof of Lemma 3.2.For fixed N ∈ N and all n ≥ N one has By definition of (α i ) and δ, the right-hand side tends to δ as n → ∞.This proves part (i).
(ii) By the ordering of the entries of λ(n), we obtain for N ∈ N and n ≥ N that Taking the limit n → ∞ on both sides, we obtain that As lim N →∞ α N = 0, this implies that δ ≤ ∞ i=1 α 2 i and therefore γ = 0.
We shall throughout fix a strictly positive multiplicity k > 0 on A n−1 and suppress it in our notation.
For sequences (λ(n)) n∈N of spectral parameters λ(n) ∈ R n with growing dimension n, we are interested in the convergence behaviour of the Bessel functions J An−1 (iλ(n), .) as n → ∞.For this, we consider J An−1 (λ, .) as a function on C r for all r ≤ n by For later use, we record the following representation.
Proposition 3.4.For λ ∈ C n and z ∈ C r with r ≤ n, with the renormalized Jack polynomials and the generalized Pochhammer symbol Together with the stability property (2.2), the assertion follows.
We shall prove the following theorem: Theorem 3.5.Let (λ(n)) n∈N be a sequence of spectral parameters λ(n) ∈ R n such that each λ(n) is decreasing with respect to ≪ .Then for fixed multiplicity k > 0, the following statements are equivalent.
(2) The sequence of Bessel functions J An−1 (iλ(n), .) n∈N converges locally uniformly on compact subsets of R (∞) , i.e. the convergence is locally uniform on each of the spaces R r , r ∈ N.
(3) For each fixed multi-index of length r, the corresponding coefficients in the Taylor of expansion of J An−1 (iλ(n), .) around 0 ∈ R r converge as n → ∞.
(4) For all symmetric functions f : R (∞) → C, the limit Moreover, in this case one has where (α = (α i ), β, γ) are the VK parameters of the VK sequence (λ(n)) n∈N , and the product on the right side extends analytically to R r for each r ∈ N.
Remark 3.6.In the geometric case k = 1, i.e. for Hermitian matrices over C, this result essentially goes back to [OV96], while in [Bo07], where also F = R and H are considered, only the limit (3.3) is established, by completely different methods and under the additional condition γ = 0.
Our proof of Theorem 3.5 is inspired by the methods of [OV96], [OO98] and [Fa08, Chapter 3].We start with the following observation.
Theorem 3.7.Assume that (λ(n)) n∈N is a VK sequence with parameters ω = (α, β, γ).Then where the series in the last case is absolutely convergent.In particular, for each symmetric function f on R (∞) , the limit Proof.We only have to consider the case m ≥ 3.In view of the ordering of λ(n) we have for arbitrary N ∈ N that The expression on the right side converges to α m−2 N δ as n → ∞.As α is squaresummable by Lemma 3.2, this implies that for each ǫ > 0, there exists an index N ∈ N such that for all n ∈ N, By the definition of a VK sequence, the last sum tends to zero as n → ∞.As ǫ > 0 was arbitrary, this finishes the proof.
We next consider for λ ∈ C (∞) the complex function where ζ → ζ k denotes the principal holomorphic branch of the power function on C\] − ∞, 0].For fixed λ, the product is finite and Φ(λ; .) is holomorphic in a neighborhood of 0 in C. According to formula (2.9) of [OO98], where m l := #{r ∈ N : i r = l} denotes the multiplicity of the number l in the tuple (i 1 , . . ., i j ) and (k Moreover, from [OO98, formula (2.8)] and the connection between the C-and P -normalizations of the Jack polynomials according to formula (12.135) of [Fo10], one calculates that (3.7) (For partitions κ = (j) with just one part, the Jack polynomials C (j) and P (j) coincide).

Proof. (1) Power series expansion around
with some constant C δ > 0. Recall that α is decreasing w.r.t.≪ and squaresummable.Hence for fixed n ∈ N, the product is holomorphic in S and even in S if α l ≥ 0 for all l.As lim l→∞ α l = 0, it follows that ψ(ω; .) is holomorphic in S or even in S. Unless α is identical zero (which is equivalent to α 1 = 0), Ψ(ω; .) has a singularity in z = 1 α1 .
(1) Since both sides of (3.8) have value 1 in z = 0, it suffices to verify that they have the same logarithmic derivative.Let log be the principle holomorphic branch of the logarithm in C\] − ∞, 0]. for |z| small enough, This is exactly the logarithmic derivative of the right-hand side in (3.8).
(2) For the second assertion, note that for m ≥ 2 we may estimate n 2 with a constant C m > 0 independent of n, which follows from (3.4).Since the righthand side converges for n → ∞, the sequence on the left-hand side is uniformly bounded in n.Hence there exists some ǫ > 0 such that for each n, the series converges for |z| < ǫ, and the dominated convergence theorem shows that (h n ) converges for n → ∞ to locally uniformly in {z ∈ C : |z| < ǫ}.Now consider Ψ(ω; .), which is holomorphic in a neighborhood of 0. Taking the logarithmic derivative as in the proof of [Fa08, Prop.3.12] and recalling Theorem 3.7, we obtain The right-hand side in equation (3.9) has the same logarithmic derivative.Since Φ λ(n) n ; 0 = 1 = Ψ(ω; 0), this proves the stated limit.We now consider the asymptotic behaviour of the Bessel functions J An−1 as n → ∞.We shall employ the following useful convergence property, see [Fa08,Propos. 3.11]: Lemma 3.10.Let (ϕ n ) n∈N be a sequence of smooth, positive-definite functions on R r with ϕ n (0) = 1 and ϕ an analytic function in some neighborhood of 0 ∈ R r .Assume that lim n→∞ ∂ α ϕ n (0) = ∂ α ϕ(0) for all α ∈ N r 0 , i.e. the Taylor coefficients of ϕ n around 0 converge to those of ϕ.Then ϕ has an analytic extension to R r , and the sequence (ϕ n ) converges to ϕ locally uniformly on R r .
which is actually a finite product.
Theorem 3.11.Assume that (λ(n)) n∈N is a VK sequence with parameters ω = (α, β, γ).Then for x ∈ R (∞) , the Bessel functions of type A n−1 with multiplicity k > 0 satisfy Proof.The Cauchy identity for Jack polynomials, see for instance [St89, Prop.2.1], states for λ ∈ C (∞) and z ∈ C r with |z j | small enough that Thus by Proposition 3.9, where the convergence is locally uniform in z in some open neighborhood of 0 ∈ C r .Therefore, the coefficients in the power series expansion of the left side around z = 0 must converge (as n → ∞) to the corresponding coefficients of ϕ.Moreover, by Lemma 3.8 we know that ϕ extends analytically to R r , and by Theorem 3.7 we have It therefore follows for all z in some neighborhood of 0 ∈ C r that Now consider the functions which are positive definite on R r for r ≤ n.In view of Proposition 3.4, So we are able to apply Lemma 3.10 to obtain that ϕ n → ϕ locally uniformly on R r .
Remark 3.12.The proof shows that for z ∈ C r with |z| small enough, Lemma 3.13.Consider a sequence (λ(n)) n∈N such that each λ(n) ∈ R n is decreasing with respect to ≪ .Suppose that the sequence of Bessel functions J An−1 (iλ(n), .) converges pointwise on R to a function which is continuous at 0. Then (λ(n)) is a VK sequence.
Proof.Put ϕ n (x) := J An−1 (iλ(n), x), x ∈ R and ϕ(x) := lim n→∞ ϕ n (x).In view of representation (2.1), there exist compactly supported probability measures µ n on R such that By Lévy's continuity theorem, there exists a probability measure µ on R such that µ n → µ weakly and In particular, the family of measures {µ n : n ∈ N} is tight.Recall the functions g j (λ) from (3.6).By Proposition 3.4 and formula (3.7) we have Hence the moments of the measures µ n are given by R ξ j dµ n (ξ) = j! g j (λ(n)) (kn) j .
We now employ Lemma 5.2 of [OO98].From the definition of the functions g j one can find a constant C > 0 such that g 4 (λ) ≤ Cg 2 (λ) 2 for all λ ∈ R (∞) , which shows that the quotient 2 is bounded as a function of n ∈ N. Hence we conclude from Lemma 5.1.of [OO98] that the sequence R ξ 2 dµ n (ξ) n∈N is bounded, which in turn implies that the are bounded as well.Standard compactness arguments and a diagonalization argument imply that (λ(n)) n∈N has a subsequence which is Vershik-Kerov.Finally, consider two such subsequences (λ l (n)) n∈N with VK parameters ω l , l = 1, 2. Then by Theorem 3.11 and our assumptions, Hence Ψ(ω 1 ; .) = Ψ(ω 2 ; .), and Proposition 3.8 implies that ω 1 = ω 2 .It follows that the full sequence (λ(n)) n∈N is Vershik-Kerov.
Putting things together, we are now able to finalize the proof of Theorem 3.5.
We shall now parametrize the possible limit functions in Theorem 3.11.We put Note that for (α, β, γ) ∈ Ω, either all entries of α are non-zero, or all entries up to finitely many are zero.
Proof.We divide the proof into several steps.(i) Assume that α = 0. Then for arbitrary ǫ > 0, there exists a sequence x = (x i ) i∈N in R such that To see this, choose N ∈ N such that 6γ π 2 N 1/2 ≤ ǫ and start with the alternating sequence It satisfies the first and the third condition of (3.13), and by the Riemann rearrangement theorem, there exists a rearrangement (x i ) i∈N of (x ′ i ) i∈N satisfying the second condition as well.For each m ∈ N we can therefore find a real sequence We may also assume that n m+1 > n m for all m.Rearranging the entries of each tuple (x nm ) according to ≪, we thus obtain a sequence (λ(n m ) ′ ) m∈N where each λ(n m ) ′ ∈ R nm is decreasing w.r.t.≪ and satisfies lim m→∞ λ(n m ) ′ i = 0 for all i ∈ N, Finally, put λ(n m ) := n m λ(n m ) ′ and λ(n) := (nλ(n m ) ′ , 0, . . ., 0) ∈ R n for n m < n < n m+1 .Then (λ(n)) n≥n1 is a VK sequence with parameters (α = 0, β, γ).
Together with Lemma 3.8, this result shows that the possible limits (for n → ∞) of the Bessel functions J An−1 (iλ(n), x) with x ∈ R r and λ(n) ∈ R n are exactly all the infinite products Ψ ω; ix k , of Theorem 3.11, which are in bijective correspondence with the parameters ω ∈ Ω.
Let us finally come back to the Olshanski spherical pair (G ∞ , K ∞ ) as in (3.14).From our results, we obtain the following result of Pickrell [Pi91] mentioned at the beginning of this section: Corollary 3.15.The set of positive definite spherical functions of the Olshanski spherical pair Proof.For a topological group H consider the set and denote by ex(P 1 (H)) the set of its extremal points.In [Ol90,Theorem 22.10] it is proven that each ϕ ∈ ex(P 1 (G ∞ )) can be approximated locally uniformly by a sequence of functions ϕ n ∈ ex(P 1 (G n )).An inspection of the proof shows that this statement remains true for biinvariant functions, i.e. each K ∞ -biinvariant ϕ ∈ ex(P 1 (G ∞ )) can be approximated locally uniformly by a sequence of K nbiinvariant functions ϕ n ∈ ex(P 1 (G n )).According to [Ol90, Theorems 23.3 and 23.6], the positive definite spherical functions of a spherical pair (G, K) (an Olshanski spherical pair or a Gelfand pair) are exactly those elements of ex(P 1 (G)) which are K-biinvariant.Thus, for a positive definite spherical function ϕ of (G ∞ , K ∞ ), there exists a sequence (ϕ n ) of positive definite spherical functions of (G n , K n ) which converges locally uniformly to ϕ. (This is also noted in [OV96, Theorem 3.5]).By Lemma 2.1, ϕ n is given by a positive definite Bessel function with some real spectral parameter λ(n) ∈ R n .Without loss of generality we may assume that λ(n) is decreasing w.r.t ≪ .From Theorem 3.5 it now follows that (λ(n)) has to be a VK sequence and that ϕ = ϕ ω , where ω are the VK parameters of (λ(n)).Conversely, starting with ω ∈ Ω we may choose an associated VK sequence (λ(n)) by Proposition 3.14.Then (3.14) defines a sequence (ϕ n ) of positive definite spherical functions of (G n , K n ) which converge to ϕ ω locally uniformly according to Theorem 3.5.It is then clear from the definitions that ϕ ω is a positive definite Olshanski-spherical function of (G ∞ , K ∞ ).

The type B case
Recall from Section 2 the Bessel functions J Bn of type B n .As n → ∞, we shall consider them with the multiplicities κ n := (k ′ n , k) with value k > 0 on the roots ±(e i ± e j ) and k ′ n ≥ 0 on the roots ±e i .It will become clear at the end of this section why the multiplicity parameter k n is allowed to vary with n.With where the Jack polynomials are of index 1/k.Recall the stability property (2.2) of the Jack polynomials.Adopting the notation from (3.2), we therefore have for Moreover, by relation (3.10), As in the proof of Theorem 3.11, we conclude that the Taylor coefficients of ϕ n around 0 converge to those of This function is actually real-analytic on R r by Lemma 3.8, since the entries of α are non-negative.As the functions ϕ n are positive definite on R n , Lemma 3.10 yields the assertion.
Proof.The proof is similar to that of Lemma 3.13.For x ∈ R, put with certain compactly supported probability measures µ n on R. By the symmetry properties of J Bn , the measure µ n is even, hence its odd moments vanish.Let further ϕ(x) := lim n→∞ ϕ n (x).Again by Lévy's continuity theorem, there exists a probability measure µ on R such that µ n → µ weakly and ϕ(x) = R e ixξ dµ(ξ) for all x ∈ R.
Further, the family {µ n : n ∈ N} is tight.From (4.1) and formula (3.7) we deduce that This shows that the even moments of µ n are given by As in the proof of Lemma 3.13, we deduce that the quotient Proof of Theorem 4.1.From Theorem 3.5 it is clear that the statements (1) and (4) are equivalent.The equivalence of (3) and (4) follows from expansion (4.2) and the fact that the Jack polynomials span the algebra of symmetric functions.By Lemma 4.4, statement (2) implies (1).Finally, Lemma 4.3 shows that statement (1) implies statement (2).
We finally want to determine the set of all parameters ω = (α, β, 0) which occur as VK parameters of a non-negative Vershik-Kerov sequence as in Theorem 4.1.Recall that in the non-negative case, the parameter γ is automatically zero by Lemma 3.2.
Proposition 4.5.The set Ω + of all pairs (α, β) for which there exists a nonnegative VK sequence with parameters (α, β, 0) is given by Proof. 1.If (α, β, 0) are the VK parameters of a VK sequence (λ(n)) with λ(n) i ≥ 0 for all i, then obviously β ≥ 0 and α 1 ≥ α 2 ≥ . . .≥ 0.Moreover, for fixed N ∈ N and n ≥ N we have As n → ∞, the first sum tends to 0 and the second sum tends to β.This proves that ∞ i=1 α i ≤ β. 2. Conversely, let (α, β) ∈ Ω + .In order to construct an associated non-negative VK sequence, we proceed in two steps.(i) Assume that α has at most finitely many non-zero entries.If α = 0, let m ∈ N be maximal such that α i = 0 for i ≤ m.If α = 0, let m := 0. Put Note that the entries of λ(n) are non-negative and decreasing for n large enough, say n ≥ n 0 .It is now straightforward to verify that (λ(n)) n≥n0 is a VK sequence with parameters (α, β, 0).(ii) Assume that all entries of α are strictly positive.Then a diagonalization argument as in the proof of Proposition 3.14 shows that there exists a VK sequence with parameters (α, β, 0).
Let us finally turn to consequences in the geometric cases, related to the Cartan motion groups of non-compact Grassmann manifolds.
For strictly increasing sequences of dimensions (p n ) n∈N , (q n ) n∈N with p n ≥ q n consider the sequence of Gelfand pairs (G n , K n ) with G n = (U pn (F) × U qn (F)) ⋉ M pn,qn (F), K n = U pn (F) × U qn (F) (4.3) over F = R, C, H.It is easily checked that the associated Olshanski spherical pair (G ∞ , K ∞ ) is independent of the specific choice of the sequences (p n ), (q n ), and so the same holds for its spherical functions.Recall from Section 2 that the positive definite spherical functions of (G n , K n ), considered as functions on the chamber C qn ⊂ R qn , are given by the Bessel functions ϕ λn (x) = J Bq n (κ n , iλ n , x), λ n ∈ R qn with κ n = (k ′ n , k) = d 2 (p n − q n + 1) − 1 2 , d 2 .Corollary 4.6.
Proof.For part (1), choose (G n , K n ) with q n = n.The proof is then the same as that of Corollary 3.15 in the type A case.Part (2) is then immediate from Theorem 3.5 and Remark 4.2.
Remark 4.7.We mention that for F = C, part (1) of this corollary is in accordance with results of [Bo19], where for the semigroup Herm + ∞ (C) of infinite dimensional positive definite matrices over C, the positive definite Olshanski spherical functions of (U ∞ (C) ⋉ Herm + ∞ (C), U ∞ (C)) were determined by semigroup methods and a reduction to the type A case.
Now we conclude from [OO98, Lemma 5.2] (employing the Lemma for the image measure of µ n under ξ → ξ 2 ) that the sequence R ξ 4 dµ n (ξ) is bounded.As (ν n ) 2 ∼ ν 2 n and (kn) 2 ∼ (kn) 2 for n → ∞, it follows that the sequence n∈N is bounded as well.Continuing as in the proof of Lemma 3.13 we obtain that λ(n) 2 νn is a Vershik-Kerov sequence.