On the space of 𝐾-finite solutions to intertwining differential operators

Kubo, Toshihisa; Ørsted, Bent

doi:10.1090/ert/527

Cited by 5 publications

(15 citation statements)

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“…where R denotes the infinitesimal right translation and X, Y are certain nilpotent elements in sl(3, R) (see (3.1)). The differential operator s for m = 3 in (1.1) in particular recovers the second order differential operator studied in [27] as the case of s = 0. Some algebraic and analytic properties of the Heisenberg ultrahyperbolic operator s as well as its generalizations are investigated in [17,18,19,20] for a linear group SL(m, R).…”

Section: Introductionmentioning

confidence: 67%

“…Lastly, the K-type formulas Sol(0; σ) K for the case of s = 0 are previously classified by ourselves in [27,Thm. 1.6].…”

Section: Table 1 Comparison Between [18] and This Paper Formentioning

confidence: 99%

“…Hypergeometric and Heun's differential equations. The main idea for accomplishing Theorem 1.7 is use of the decomposition formula (1.5) and a refinement of the method applied for the case s = 0 in [27]. The central problem is classifying the space Sol(s; n) of K-type solutions.…”

Section: Table 1 Comparison Between [18] and This Paper Formentioning

confidence: 99%

“…This paper consists of ten sections including this introduction. In Section 2, we overview a general framework established in [27] for the Peter-Weyl theorem (1.5) for the space of K-finite solutions to intertwining differential operators.…”

Section: Table 1 Comparison Between [18] and This Paper Formentioning

confidence: 99%

“…In this paper, we consider G = SL(3, R), the universal covering group of SL(3, R). By making use of a technique from our earlier paper [27], we successfully classified the K-type formulas of Sol( s ) K for SL(3, R). Our method is different from the algebraic theory used in [18]; we rather utilize differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Classification of $K$-type formulas for the Heisenberg ultrahyperbolic operator $\square_s$ for $\widetilde{SL}(3,\mathbb{R})$ and tridiagonal determinants for local Heun functions

Kubo,

Ørsted

2021

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The K-type formulas of the space of K-finite solutions to the Heisenberg ultrahyperbolic equation sf = 0 for the non-linear group SL(3, R) are classified. This completes a previous study of Kable for the linear group SL(m, R) in the case of m = 3, as well as generalizes our earlier results on a certain second order differential operator. As a by-product we also show several properties of certain sequences {Pj (x; y)} ∞ j=0 and {Qj(x; y)} ∞ j=0 of tridiagonal determinants, whose generating functions are given by local Heun functions. In particular, it is shown that these sequences satisfy a certain arithmetic-combinatorial property, which we refer to as a palindromic property. We further show that classical sequences of Cayley continuants {Cay j (x; y)} ∞ j=0 and Krawtchouk polynomials {Kj (x; y)} ∞ j=0 also admit this property. In the end a new proof of Sylvester's formula for certain tridiagonal determinant Sylv(x; n) is provided from a representation theory point of view.

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Section: Introductionmentioning

confidence: 67%

“…Lastly, the K-type formulas Sol(0; σ) K for the case of s = 0 are previously classified by ourselves in [27,Thm. 1.6].…”

Section: Table 1 Comparison Between [18] and This Paper Formentioning

confidence: 99%

Section: Table 1 Comparison Between [18] and This Paper Formentioning

confidence: 99%

Section: Table 1 Comparison Between [18] and This Paper Formentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%