Affine Pieri rule for periodic Macdonald spherical functions of the excluded twisted type $\widehat{BC}_n=A^{(2)}_{2n}$, and identification of the deformed fusion ring with deformed Verlinde algebras

Van Diejen, Emsiz, and Zurrián established an affine Pieri rule for periodic Macdonald spherical functions associated with any untwisted affine Lie algebra or its twisted counterpart, with exactly one family excluded: the twisted affine Lie algebras of type $\widehat{BC}_n=A^{(2)}_{2n}$, the unique Kac family whose finite (gradient) root system is non-reduced. The open problem is to complete their program: construct the periodic (level-$c$) Macdonald spherical functions of Koornwinder type attached to $A^{(2)}_{2n}$, prove an affine Pieri rule — an explicit formula for the structure constants of multiplication by minuscule and quasi-minuscule basis elements — and thereby complete the deformed genus-zero Wess–Zumino–Witten fusion ring to all affine Lie types; and to settle the source's stated expectation that these deformed fusion rings are isomorphic to the deformed Verlinde algebras of Teleman and Teleman–Woodward. No resolution of these two questions is known in the literature to date.

Background

A genus-zero Wess–Zumino–Witten (WZW) conformal field theory based on an affine Lie algebra $\hat{\mathfrak{g}}$ at positive integral level $c$ has a fusion ring (also called the Verlinde algebra): a finite-dimensional commutative algebra whose structure constants encode which representations may fuse to which, with a natural basis indexed by the dominant weights of level at most $c$. In type $\widehat{sl}(n)_c$ this ring is realized by symmetric polynomials evaluated on a finite grid: at the Schur level ($t=0$) the structure constants are quantum Littlewood–Richardson coefficients, which compute genus-zero Gromov–Witten invariants of Grassmannians. A Pieri rule is an explicit formula for multiplication by a distinguished generator; for Hall–Littlewood polynomials (a one-parameter $t$-deformation of Schur polynomials) it dates back to Macdonald's book, and Korff extended it to an affine Pieri rule for cylindric Hall–Littlewood polynomials — Hall–Littlewood polynomials labeled by partitions on a periodic (cylindric) lattice — thereby describing a $t$-deformation of the $\widehat{sl}(n)_c$ fusion ring (a deformed Verlinde algebra).

For simple Lie algebras of arbitrary type, the role of Hall–Littlewood polynomials is played by Macdonald spherical functions, originally constructed as spherical functions on $p$-adic symmetric spaces; they are two-parameter ($q,t$) deformations of the characters and include the Hall–Littlewood polynomials as a specialization. Van Diejen and Emsiz proved a Pieri formula for them in arbitrary simple Lie type via representations of the affine Hecke algebra (the deformation algebra controlling the Weyl-group symmetry of these functions). Van Diejen, Emsiz, and Zurrián then implemented periodic boundary conditions to obtain periodic Macdonald spherical functions at level $c$, together with an affine Pieri rule: an explicit formula for the structure constants of the deformed genus-zero $\hat{\mathfrak{g}}$-fusion ring in the periodic spherical basis, for multiplication by basis elements attached to minuscule weights (dominant weights $\mu$ with $\langle\mu,\alpha^{\vee}\rangle\in\{0,1\}$ for every positive coroot) and quasi-minuscule weights (the highest short root). In type $A^{(1)}_{n-1}$ their formula reproduces Korff's cylindric Hall–Littlewood Pieri rule, which at $t=0$ specializes to the classical fusion-ring Pieri formula.

The theorem of van Diejen–Emsiz–Zurrián covers all untwisted affine Lie algebras and all twisted ones except a single family: type $\widehat{BC}_n=A^{(2)}_{2n}$, the unique twisted affine family whose gradient (finite) root system is the non-reduced system $BC_n$. The exclusion is structural: the affine Hecke algebra machinery used in the proof presupposes a reduced gradient root system, while for the non-reduced $BC_n$ case the finite-type spherical functions are Koornwinder polynomials (the $BC_n$-analogue of Macdonald polynomials) governed by a different, larger Hecke-type algebra. A partial $BC_n$ result exists by other means — van Diejen deformed the $\widehat{BC}_n$ fusion ring through the Bethe ansatz diagonalization of an open $q$-boson model with diagonal boundary conditions — but that construction does not use affine Hecke algebras, does not produce a periodic Macdonald spherical basis, and does not yield a Pieri rule with explicit structure constants of the kind established for the other types. In the same paper the authors also state an expectation that goes beyond the type coverage: the $q,t$-deformed fusion rings they construct should be isomorphic to the deformed Verlinde algebras arising in Teleman's and Teleman–Woodward's work on $K$-theory of moduli spaces of bundles; at the time this was manifest only for $\widehat{sl}(2)_c$. Both questions — the excluded non-reduced type and the deformed-Verlinde identification — remain open.

Problem Statement

Extend the affine Pieri rule of van Diejen–Emsiz–Zurrián to the one excluded case, and settle the identification they conjectured. Concretely: (1) for the twisted affine Lie algebra of type $\widehat{BC}_n=A^{(2)}_{2n}$ — equivalently, for the non-reduced situation whose finite-type spherical functions are the Koornwinder ($BC_n$) analogues of Macdonald spherical functions — construct, for every rank $n\geq 1$ and every integer level $c>1$ (the level range of the source's framework), the periodic Macdonald spherical functions as a finite basis of a deformed genus-zero Wess–Zumino–Witten fusion ring, and prove an affine Pieri rule for them: an explicit formula for the structure constants of multiplication by basis elements attached to minuscule and quasi-minuscule weights, valid uniformly in the deformation parameters — for the non-reduced case, the full Koornwinder-type parameter family (a larger family than the source's root-multiplicity function $t$ with values in $(-1,1)\setminus\{0\}$) — with the $t=0$ specialization recovering the undeformed $\widehat{BC}_n$ fusion-ring Pieri formula; and (2) prove or disprove the source's stated expectation that the deformed fusion rings (for the types already covered by the source, and for the new type if part (1) succeeds) are isomorphic to the deformed Verlinde algebras of Teleman and Teleman–Woodward. These two deliverables jointly complete one overarching objective: to pin down the $q,t$-deformed genus-zero fusion ring for every affine Lie algebra, both its structure constants (via the Pieri rule) and its intrinsic geometric/representation-theoretic meaning (via the Verlinde identification).

References

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