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Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

Evgeny Feigin ORCID logo, Syu Kato ORCID logo and Ievgen Makedonskyi

Abstract

We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules for simply-laced Dynkin types except for type E8.

Funding statement: The research of E. Feigin is supported by the grant RSF 19-11-00056. The research of I. Makedonskyi is supported by the grant RSF 19-11-00056 and JSPS Postdoctoral Fellowships for Research in Japan JP18F18014. The research of S. Kato is supported in part by JSPS Grant-in-Aid for Scientific Research (B) JP26287004 and JP19H01782.

A Numerical equality

We discuss the equality of Theorem 5.12 on the level of characters. To this end, we continue to assume that 𝔤 is of type ADE. Except for this, we generally follow the setting of Section 2.

Consider the Cherednik kernel

κ(x,q,t)=αΔ+a(1-eα)multααΔ+a(1-teα)multα[P]((q))[[t]]=[x1±1,,xn±1]((q))[[t]]

through the identifications eωi=xi for 1in and q=eδ.

We consider the Euler–Poincaré pairing

[𝔅]×[𝔅0](M,N)(M,N)EP:=i=0(-1)i𝗀𝖽𝗂𝗆ext𝔅i(M,N*)*,

as the formal sum. Here a certain category 𝔅 of -modules and its full subcategories 𝔅 and 𝔅0 are borrowed from Section 5. This pairing lands in ((q)).

The Euler–Poincaré pairing satisfies the following properties:

  1. (i)

    It is q-linear.

  2. (ii)

    For a short exact sequence:

    0M1MM20,

    we have

    (M,N)EP=(M1,N)EP+(M2,N)EP

    and the same equality holds for a short exact sequence in the second argument. Thus the Euler–Poincaré pairing depends only on the characters of M and N.

  3. (iii)

    We have the following equality:

    (PΛ,Γ)EP=δΛ,-Γ,Λ,ΓPa.
  4. (iv)

    If both M and N belong to 𝔅0, then we have

    (M,N)EP=(N,M)EP.

The proofs of these properties are standard and are thus omitted (the last item requires [15, Sections 2.1–2.2] as in Section 5).

The properties (i), (ii), (iii) completely characterizes the Euler–Poincaré pairing. Now consider the specialization of the Cherednik inner product on [x±1]((q)):

(P(x,q),Q(x,q))C:=(P(x,q)Q(X,q)κ(x,q,0))0,

where the lower index 0 denotes the constant term with respect to q in the power series expansion of h. Applying (i)–(iii) repeatedly, we obtain

(M,N)EP=(chM,chN)C.

Recall that we have defined graded -modules Dλ and Uλ for each λP in Section 3.1 such that we have

(A.1)chDλ=Eλ(x,q,0)andchU-λ=Eλ(x-1,q-1,)

by the non-symmetric Macdonald polynomial Eλ(x,q,t) (see Theorem 5.1 and Remark 3.28).

Theorem A.1.

For each λ,μP we have

(chUμ,chDλ)C=δλ+μ,0(q)(λ-)σ=(Uμ,Dλ)EP,

where σW is the shortest element such that λ=σλ- for some λ-P-, and (λ-)σ is defined in Section 3.2.

Proof.

For f(x,q,t)[P](q,t), we set

f(x,q,t)¯=f(x-1,q-1,t-1).

By the definition of the non-symmetric Macdonald polynomials (see e.g. [7]), we have

(Eλ(x,q,t)Eμ(x,q,t)¯κ(x,q,t))0=0

for λμ. In other words,

(Eλ(x,q,t)Eμ(x-1,q-1,t-1)κ(x,q,t))0=0.

Substituting t0, we obtain

(chDλ,chU-μ)C=0

by (A.1). By [8, (3.4.2)], setting μ=λ yields

(Eλ(x,q,t)Eλ(x-1,q-1,t-1)κ(x,q,t))0|t0=(q)(λ-)σ,

and we obtain the needed relation by the same argument. ∎

Acknowledgements

We benefited from the SageMath, Nonsymmetric Macdonald polynomials package by A. Schilling and N. M. Thiery (2013).[1]

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Received: 2017-12-13
Revised: 2019-03-26
Published Online: 2019-06-13
Published in Print: 2020-07-01

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