Correction factors for Kac–Moody groups and t-deformed root multiplicities

Abstract

We study a correction factor for Kac–Moody root systems which arises in the theory of p-adic Kac–Moody groups. In affine type, this factor is known, and its explicit computation is the content of the Macdonald constant term conjecture. The data of the correction factor can be encoded as a collection of polynomials \(m_\lambda \in \mathbb {Z}[t]\) indexed by positive imaginary roots \(\lambda \). At \(t=0\) these polynomials evaluate to the root multiplicities, so we consider \(m_\lambda \) to be a t-deformation of \({{\,\mathrm{mult}\,}}(\lambda )\). We generalize the Peterson algorithm and the Berman–Moody formula for root multiplicities to compute \(m_\lambda \). As a consequence we deduce fundamental properties of \(m_\lambda \).

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Acknowledgements

We thank Alexander Braverman, Paul E. Gunnells, Kyu-Hwan Lee, Dongwen Liu, Peter McNamara, Manish Patnaik for helpful conversations. At the beginning of this project the first author was partially supported by a PIMS postdoctoral fellowship and the second author was supported through Manish Patnaik’s Subbarao Professorship in Number Theory and an NSERC Discovery Grant at the University of Alberta. The project started at the workshop “Whittaker functions: Number Theory, Geometry and Physics” at the Banff International Research Station in 2016; we thank the organizers of this workshop. We also thank the anonymous referee for helpful comments.

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Appendix A. Examples of the polynomial \(\chi _{\lambda }\)

Appendix A. Examples of the polynomial \(\chi _{\lambda }\)

At the end of Sect. 7 we set \(\chi _\lambda (t) = \frac{m_\lambda (t)}{(1-t)^2}\) for imaginary roots \(\lambda \). Theorem 7.6 tells us that \(\chi _\lambda \in \mathbb {Z}[t]\). In this section we tabulate certain polynomials \(\chi _\lambda (t)\), computed with the algorithm of Sect. 5.

For the rank 2 hyperbolic root system with Cartan matrix \(\begin{bmatrix}2&-3 \\-2&2 \\\end{bmatrix}\), we have:

$$\begin{aligned} \begin{array}{lll} \lambda &{} \chi _{\lambda } \\ \left( 1, 1\right) &{} 1 \\ \left( 2, 2\right) &{} -t + 1 \\ \left( 3, 2\right) &{} t^{2} + 2 \\ \left( 3, 3\right) &{} -t^{3} - 2 t + 2 \\ \left( 4, 3\right) &{} t^{4} - t^{3} + 2 t^{2} - 3 t + 3 \\ \left( 4, 4\right) &{} -t^{5} + t^{4} - 2 t^{3} + 3 t^{2} - 6 t + 3 \\ \left( 5, 4\right) &{} t^{6} - 2 t^{5} + 4 t^{4} - 6 t^{3} + 9 t^{2} - 9 t + 6 \\ \left( 5, 5\right) &{} -t^{7} + t^{6} - 4 t^{5} + 6 t^{4} - 10 t^{3} + 13 t^{2} - 13 t + 7 \\ \left( 6, 4\right) &{} t^{6} - 4 t^{5} + 5 t^{4} - 8 t^{3} + 11 t^{2} - 13 t + 6 \\ \cdots &{} \cdots \\ \left( 10, 9\right) &{} t^{16} - 7 t^{15} + 29 t^{14} - 91 t^{13} + 248 t^{12} - 584 t^{11} + 1197 t^{10} - 2170 t^{9} + 3505 t^{8} - 5039 t^{7} + 6437 t^{6} - \\ &{}7253 t^{5} + 7042 t^{4} - 5618 t^{3} + 3405 t^{2} - 1372 t + 272\\ \end{array} \end{aligned}$$

For the symmetric rank 2 hyperbolic root system with Cartan matrix \(\begin{bmatrix}2&-3 \\-3&2 \\\end{bmatrix}\):

$$\begin{aligned} \begin{array}{lll} \lambda &{} \chi _{\lambda } \\ \left( 1, 1\right) &{} 1 \\ \left( 2, 2\right) &{} -2 t + 1 \\ \left( 2, 3\right) &{} t^{2} - t + 2 \\ \left( 3, 2\right) &{} t^{2} - t + 2 \\ \left( 3, 3\right) &{} -2 t^{3} + 3 t^{2} - 4 t + 3 \\ \left( 3, 4\right) &{} t^{4} - 3 t^{3} + 6 t^{2} - 6 t + 4 \\ \left( 4, 3\right) &{} t^{4} - 3 t^{3} + 6 t^{2} - 6 t + 4 \\ \left( 4, 4\right) &{} -2 t^{5} + 7 t^{4} - 12 t^{3} + 17 t^{2} - 16 t + 6 \\ \left( 4, 5\right) &{} t^{6} - 5 t^{5} + 15 t^{4} - 26 t^{3} + 30 t^{2} - 23 t + 9 \\ \left( 4, 6\right) &{} t^{6} - 8 t^{5} + 19 t^{4} - 31 t^{3} + 36 t^{2} - 28 t + 9 \\ \left( 5, 4\right) &{} t^{6} - 5 t^{5} + 15 t^{4} - 26 t^{3} + 30 t^{2} - 23 t + 9 \\ \left( 5, 5\right) &{} -2 t^{7} + 9 t^{6} - 30 t^{5} + 58 t^{4} - 82 t^{3} + 77 t^{2} - 50 t + 16 \\ \left( 6, 4\right) &{} t^{6} - 8 t^{5} + 19 t^{4} - 31 t^{3} + 36 t^{2} - 28 t + 9 \\ \cdots &{} \cdots \\ \left( 10, 9\right) &{} t^{16} - 15 t^{15} + 135 t^{14} - 811 t^{13} + 3535 t^{12} - 11729 t^{11} + 30615 t^{10} - 64282 t^{9} + 110096 t^{8} - \\ &{}154852 t^{7} + 178868 t^{6} - 168420 t^{5} + 127110 t^{4} - 74539 t^{3} + 32094 t^{2} - 9070 t + 1267 \\ \end{array} \end{aligned}$$

For the Feingold–Frenkel rank 3 hyperbolic root system with Cartan matrix \(\begin{bmatrix}2&-2&0 \\ -2&2&-1 \\ 0&-1&2 \end{bmatrix}\):

$$\begin{aligned} \begin{array}{lll} \lambda &{} \chi _{\lambda } \\ \left( 1, 1, 0\right) &{}~~ 1 \\ \left( 2, 2, 0\right) &{}~~ 1 \\ \left( 2, 2, 1\right) &{}~~ 2 \\ \left( 3, 3, 0\right) &{}~~ 1 \\ \left( 3, 3, 1\right) &{}~~ -t + 3 \\ \left( 3, 4, 2\right) &{}~~ -2 t + 5 \\ \left( 4, 4, 0\right) &{}~~ 1 \\ \left( 4, 4, 1\right) &{}~~ -2 t + 5 \\ \left( 4, 4, 2\right) &{}~~ -t^{2} - 6 t + 7 \\ \left( 4, 5, 2\right) &{}~~ t^{3} + t^{2} - 9 t + 11 \\ \left( 5, 5, 0\right) &{}~~ 1 \\ \left( 5, 5, 1\right) &{}~~ -5 t + 7 \\ \left( 5, 5, 2\right) &{}~~ 2 t^{3} + 2 t^{2} - 17 t + 15 \\ \left( 5, 6, 2\right) &{}~~ -t^{4} + 3 t^{3} + 6 t^{2} - 26 t + 22 \\ \left( 5, 6, 3\right) &{}~~ -3 t^{4} + 6 t^{3} + 13 t^{2} - 43 t + 30 \\ \left( 6, 6, 0\right) &{}~~ 1 \\ \left( 6, 6, 1\right) &{}~~ t^{2} - 8 t + 11 \\ \left( 6, 6, 2\right) &{}~~ -2 t^{4} + 5 t^{3} + 11 t^{2} - 43 t + 30 \\ \left( 6, 6, 3\right) &{}~~ -6 t^{4} + 8 t^{3} + 23 t^{2} - 65 t + 42 \\ \left( 6, 7, 2\right) &{}~~ -5 t^{4} + 6 t^{3} + 22 t^{2} - 63 t + 42 \\ \left( 7, 7, 0\right) &{}~~ 1 \\ \left( 7, 7, 1\right) &{}~~ 2 t^{2} - 15 t + 15 \\ \end{array} \end{aligned}$$

Notice that the phenomenon of \(\chi _\lambda = 1\) for \(\lambda =(n, n, 0)\) is an instance of Proposition 7.1 and Theorem 4.2, as these are roots of the embedded affine root subsystem \(A_1^{(1)}\).

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Muthiah, D., Puskás, A. & Whitehead, I. Correction factors for Kac–Moody groups and t-deformed root multiplicities. Math. Z. 296, 127–145 (2020). https://doi.org/10.1007/s00209-019-02419-1

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