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Journal articles on the topic 'Hecke'

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Published: 4 June 2021

Last updated: 25 May 2024

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1

Hyodo, Fumitake. "A formal power series of a Hecke ring associated with the Heisenberg lie algebra over ℤp." International Journal of Number Theory 11, no. 08 (November 5, 2015): 2305–23. http://dx.doi.org/10.1142/s1793042115501055.

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This paper studies a formal power series with coefficients in a Hecke ring associated with the Heisenberg Lie algebra. We relate the series to the classical Hecke series defined by Hecke, and prove that the series has a property similar to the rationality theorem of the classical Hecke series. And then, our results recover the rationality theorem of the classical Hecke series.

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2

Orellana, R. C. "Weights of Markov traces on Hecke algebras." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 508 (March 12, 1999): 157–78. http://dx.doi.org/10.1515/crll.1999.508.157.

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Abstract We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B.

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3

Kaymak, Şule, Bilal Demır, Özden Koruoğlu, and Recep Şahin. "Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 1 (March 1, 2018): 159–68. http://dx.doi.org/10.2478/auom-2018-0010.

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Abstract Let p and q be integers such that 2 ≤ p ≤ q; p + q > 4 and let Hp,q be the generalized Hecke group associated to p and q: The generalized Hecke group Hp,q is generated by X(z) = -(z-λp)-1 and Y (z) = -(z+ λq)-1 where λp = 2 cos ≤ π/p and λq = 2 cos π/q . The extended generalized Hecke group H̅p,q is obtained by adding the reection R(z) = 1/z̅ to the generators of generalized Hecke group Hp,q: In this paper, we study the commutator subgroups of generalized Hecke groups Hp,q and extended generalized Hecke groups H̅p,q.

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4

McGerty, Kevin. "On the centre of the cyclotomic Hecke algebra of G(m, 1, 2)." Proceedings of the Edinburgh Mathematical Society 55, no. 2 (April 12, 2012): 497–506. http://dx.doi.org/10.1017/s0013091510001264.

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AbstractWe compute the centre of the cyclotomic Hecke algebra attached to G(m, 1, 2) and show that if q ≠ 1, it is equal to the image of the centre of the affine Hecke algebra Haff2. We also briefly discuss what is known about the relation between the centre of an arbitrary cyclotomic Hecke algebra and the centre of the affine Hecke algebra of type A.

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5

Lee, Kyu-Hwan. "Iwahori-Hecke Algebras of SL2 over 2-Dimensional Local Fields." Canadian Journal of Mathematics 62, no. 6 (December 14, 2010): 1310–24. http://dx.doi.org/10.4153/cjm-2010-072-x.

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AbstractIn this paper we construct an analogue of Iwahori–Hecke algebras of SL2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL2, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori–Matsumoto type relations.

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6

Hsu, Chi-Yun. "Fourier coefficients of the overconvergent generalized eigenform associated to a CM form." International Journal of Number Theory 16, no. 06 (February 11, 2020): 1185–97. http://dx.doi.org/10.1142/s1793042120500608.

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Let [Formula: see text] be a modular form with complex multiplication. If [Formula: see text] has critical slope, then Coleman’s classicality theorem implies that there is a [Formula: see text]-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as [Formula: see text]. We give a formula for the Fourier coefficients of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of [Formula: see text]-adic overconvergent forms containing [Formula: see text].

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7

Khare, Chandrashekhar, and Niccoló Ronchetti. "Derived Hecke action at p and the ordinary p -adic cohomology of arithmetic manifolds." American Journal of Mathematics 145, no. 6 (December 2023): 1631–94. http://dx.doi.org/10.1353/ajm.2023.a913294.

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Abstract: We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of reductive groups ${\rm G}(\Bbb{Q})$. This is the analog at $\ell=p$ of derived Hecke actions studied by Venkatesh in the tame case, and is the derived analog of Hida's theory for ordinary Hecke algebras. We show that properties of the derived Hecke action at $p$ are related to deep conjectures in Galois cohomology which are higher analogs of the classical Leopoldt conjecture.

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8

Clozel, Laurent, Hee Oh, and Emmanuel Ullmo. "Hecke operators and equidistribution of Hecke points." Inventiones Mathematicae 144, no. 2 (May 1, 2001): 327–51. http://dx.doi.org/10.1007/s002220100126.

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9

Choi, SoYoung, and Chang Heon Kim. "Weakly holomorphic Hecke eigenforms and Hecke eigenpolynomials." Advances in Mathematics 290 (February 2016): 144–62. http://dx.doi.org/10.1016/j.aim.2015.12.002.

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10

Ciubotaru, Dan, Eric M. Opdam, and Peter E. Trapa. "Algebraic and analytic Dirac induction for graded affine Hecke algebras." Journal of the Institute of Mathematics of Jussieu 13, no. 3 (March 13, 2013): 447–86. http://dx.doi.org/10.1017/s147474801300008x.

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AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

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11

OGIEVETSKY, O. V., and L. POULAIN D'ANDECY. "ON REPRESENTATIONS OF CYCLOTOMIC HECKE ALGEBRAS." Modern Physics Letters A 26, no. 11 (April 10, 2011): 795–803. http://dx.doi.org/10.1142/s0217732311035377.

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An approach, based on Jucys–Murphy elements, to the representation theory of cyclotomic Hecke algebras is developed. The maximality (in the cyclotomic Hecke algebra) of the set of the Jucys–Murphy elements is established. A basis of the cyclotomic Hecke algebra is suggested; this basis is used to establish the flatness of the deformation without using the representation theory.

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12

Chan, Heng Huat, and Zhi-Guo Liu. "On Certain Series of Hecke-Type." New Zealand Journal of Mathematics 48 (March 13, 2020): 1–10. http://dx.doi.org/10.53733/22.

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Around 2004, J. Lovejoy proved three Hecke-type series identities using Bailey pairs. In this article, we prove Lovejoy’s identities using transformation formulas for q-series discovered by Z.G. Liu in 2013. Some new Hecke-type series are also derived. Our approach also allows us to derive some new Hecke-type identities.

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13

Choi, SoYoung, and Chang Heon Kim. "Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms." Open Mathematics 20, no. 1 (January 1, 2022): 313–32. http://dx.doi.org/10.1515/math-2022-0009.

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Abstract Extending our previous work we construct weakly holomorphic Hecke eigenforms whose period polynomials correspond to elements in a basis consisting of odd and even Hecke eigenpolynomials induced by only cusp forms. As an application of our results, we give an explicit construction of the holomorphic parts of harmonic weak Maass forms that are good for Hecke eigenforms. Moreover, we give an explicit construction of the Hecke-equivariant map between the space of weakly holomorphic cusp forms and two copies of the spaces of cusp forms, and show that the map is compatible with the corresponding map on the spaces of period polynomials.

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14

Sahin, Recep, Taner Meral, and Özden Koruoğlu. "Power and free normal subgroups of generalized Hecke groups." Asian-European Journal of Mathematics 13, no. 04 (April 4, 2019): 2050080. http://dx.doi.org/10.1142/s1793557120500801.

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Let [Formula: see text] and [Formula: see text] be integers such that [Formula: see text] [Formula: see text] and let [Formula: see text] be generalized Hecke group associated to [Formula: see text] and [Formula: see text] Generalized Hecke group [Formula: see text] is generated by [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] In this paper, for positive integer [Formula: see text] we study the power subgroups [Formula: see text] of generalized Hecke groups [Formula: see text]. Also, we give some results about free normal subgroups of generalized Hecke groups [Formula: see text]

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15

Aubert, Anne-Marie, Paul Baum, Roger Plymen, and Maarten Solleveld. "HECKE ALGEBRAS FOR INNER FORMS OF -ADIC SPECIAL LINEAR GROUPS." Journal of the Institute of Mathematics of Jussieu 16, no. 2 (May 5, 2015): 351–419. http://dx.doi.org/10.1017/s1474748015000079.

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Let$F$be a non-Archimedean local field, and let$G^{\sharp }$be the group of$F$-rational points of an inner form of$\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of$G^{\sharp }$, via restriction from an inner form$G$of$\text{GL}_{n}(F)$.For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth$G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of$G^{\sharp }$, and the idempotent is derived from a type for$G$. We show that the Hecke algebras for Bernstein components of$G^{\sharp }$are similar to affine Hecke algebras of type$A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

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16

Kaliszewski, S., Magnus B. Landstad, and John Quigg. "Hecke C*-Algebras, Schlichting Completions and Morita Equivalence." Proceedings of the Edinburgh Mathematical Society 51, no. 3 (October 2008): 657–95. http://dx.doi.org/10.1017/s0013091506001419.

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AbstractThe Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

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17

Sahin, Recep, Özden Koruoğlu, and Sebahattin İkikardes. "On the Extended Hecke Group $\overline{H}(\lambda_5)$." Algebra Colloquium 13, no. 01 (March 2006): 17–23. http://dx.doi.org/10.1142/s1005386706000046.

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We consider the extended Hecke group [Formula: see text] generated by T(z)=-1/z, [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we study the abstract group structure of the extended Hecke group and its power subgroups [Formula: see text]. Also, we give relations between power subgroups and commutator subgroups of the extended Hecke group [Formula: see text].

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18

Rouquier, Raphaël. "Quiver Hecke Algebras and 2-Lie Algebras." Algebra Colloquium 19, no. 02 (May 3, 2012): 359–410. http://dx.doi.org/10.1142/s1005386712000247.

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We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

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19

RESSLER, WENDELL. "ON BINARY QUADRATIC FORMS AND THE HECKE GROUPS." International Journal of Number Theory 05, no. 08 (December 2009): 1401–18. http://dx.doi.org/10.1142/s1793042109002730.

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We present a reduction theory for certain binary quadratic forms with coefficients in ℤ[λ], where λ is the minimal translation in a Hecke group. We generalize from the modular group Γ(1) = PSL(2,ℤ) to the Hecke groups and make extensive use of modified negative continued fractions. We also define and characterize "reduced" and "simple" hyperbolic fixed points of the Hecke groups.

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20

Walling, Lynne H. "Hecke eigenvalues and relations for Siegel Eisenstein series of arbitrary degree, level, and character." International Journal of Number Theory 13, no. 02 (February 7, 2017): 325–70. http://dx.doi.org/10.1142/s179304211750021x.

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We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalize the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalize the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.

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21

Hái, Phùng Hô. "Hecke symmetries." Journal of Pure and Applied Algebra 152, no. 1-3 (September 2000): 109–21. http://dx.doi.org/10.1016/s0022-4049(99)00145-0.

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22

Krieg, Aloys. "Hecke algebras." Memoirs of the American Mathematical Society 87, no. 435 (1990): 0. http://dx.doi.org/10.1090/memo/0435.

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23

Takebayashi, Tadayoshi. "Double affine Hecke algebras and elliptic Hecke algebras." Journal of Algebra 253, no. 2 (July 2002): 314–49. http://dx.doi.org/10.1016/s0021-8693(02)00055-8.

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24

Reeks, Michael. "Cocenters of Hecke–Clifford and spin Hecke algebras." Journal of Algebra 476 (April 2017): 85–112. http://dx.doi.org/10.1016/j.jalgebra.2016.11.039.

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25

Yang, Guiyu, and Yanbo Li. "Standardly based algebras and 0-Hecke algebras." Journal of Algebra and Its Applications 14, no. 10 (September 2015): 1550141. http://dx.doi.org/10.1142/s0219498815501418.

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In this paper we prove that standardly based algebras are invariant under Morita equivalences. As an application, we prove 0-Hecke algebras and 0-Schur algebras are standardly based algebras. From this point of view, we give a new way to construct the simple modules of 0-Hecke algebras, and prove that the dimension of the center of a symmetric 0-Hecke algebra is not less than the number of its simple modules.

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26

Zhong, Changlong. "ON THE FORMAL AFFINE HECKE ALGEBRA." Journal of the Institute of Mathematics of Jussieu 14, no. 4 (June 9, 2014): 837–55. http://dx.doi.org/10.1017/s1474748014000188.

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We study the action of the formal affine Hecke algebra on the formal group algebra, and show that the the formal affine Hecke algebra has a basis indexed by the Weyl group as a module over the formal group algebra. We also define a concept called the normal formal group law, which we use to simplify the relations of the generators of the formal affine Demazure algebra and the formal affine Hecke algebra.

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27

Vilardi, Trevor, and Hui Xue. "Distinguishing eigenforms of level one." International Journal of Number Theory 14, no. 01 (November 21, 2017): 31–36. http://dx.doi.org/10.1142/s1793042118500033.

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Assuming the irreducibility of characteristic polynomials of Hecke operators [Formula: see text], we show that two normalized Hecke eigenforms of level one are distinguished by their second Fourier coefficients.

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28

Liu, Wille. "Knizhnik–Zamolodchikov functor for degenerate double affine Hecke algebras: algebraic theory." Representation Theory of the American Mathematical Society 26, no. 30 (August 30, 2022): 906–61. http://dx.doi.org/10.1090/ert/614.

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In this article, we define an algebraic version of the Knizhnik–Zamolodchikov (KZ) functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo–Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.

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29

Baruch, Ehud Moshe, and Soma Purkait. "Newforms of Half-integral Weight: The Minus Space Counterpart." Canadian Journal of Mathematics 72, no. 2 (October 31, 2019): 326–72. http://dx.doi.org/10.4153/s0008414x19000233.

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AbstractWe study genuine local Hecke algebras of the Iwahori type of the double cover of $\operatorname{SL}_{2}(\mathbb{Q}_{p})$ and translate the generators and relations to classical operators on the space $S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$, $M$ odd and square-free. In [9] Manickam, Ramakrishnan, and Vasudevan defined the new space of $S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$ that maps Hecke isomorphically onto the space of newforms of $S_{2k}(\unicode[STIX]{x1D6E4}_{0}(2M))$. We characterize this newspace as a common $-1$-eigenspace of a certain pair of conjugate operators that come from local Hecke algebras. We use the classical Hecke operators and relations that we obtain to give a new proof of the results in [9] and to prove our characterization result.

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30

Braverman, Alexander, and David Kazhdan. "Some Examples of Hecke Algebras for Two-Dimensional Local Fields." Nagoya Mathematical Journal 183 (2006): 57–84. http://dx.doi.org/10.1017/s0027763000009314.

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Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ. For instance our spherical Hecke algebra corresponds to the subgroup G (A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

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31

PINSKY, TALI. "Templates for geodesic flows." Ergodic Theory and Dynamical Systems 34, no. 1 (November 28, 2012): 211–35. http://dx.doi.org/10.1017/etds.2012.132.

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AbstractWe construct templates for geodesic flows on an infinite family of Hecke triangle groups. Our results generalize those of E. Ghys [Knots and dynamics. Proc. Int. Congress of Mathematicians. Vol. 1. International Congress of Mathematicians, Zürich, 2007], who constructed a template for the modular flow in the complement of the trefoil knot in $S^3$. A significant difficulty that arises in any attempt to go beyond the modular flow is the fact that for other Hecke triangles the geodesic flow cannot be viewed as a flow in $S^3$, and one is led to consider embeddings into lens spaces. Our final result is an explicit description of a single ‘Hecke template’ which contains all other templates we construct, allowing a topological study of the periodic orbits of different Hecke triangle groups all at once.

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32

Müller, Jens, and Matthias Schindler. "Funktionen von Hecken als Habitate für die Avifauna im Naturschutzgebiet "Rodderberg" bei Bonn." Decheniana : Verhandlungen des Naturhistorischen Vereins der Rheinlande und Westfalens 161 (January 1, 2008): 75–86. http://dx.doi.org/10.21248/decheniana.v161.3833.

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Im Naturschutzgebiet Rodderberg südlich von Bonn wurde die Funktion verschiedener Hecken als Brut- bzw. Nahrungshabitat für Vögel untersucht. Während der Brutzeit (April-Juli 2005) wurden an neun Hecken mit einer Gesamtlänge von 1180 m Brutreviere und das Territorialverhalten von Vögeln kartiert. Außerdem wurden verschiedene Strukturparameter (Länge, Breite, Höhe, Anzahl der Lücken, Überhälter, Heckendichte, Abstand zu Waldbiotopen und Siedlungsflächen) zur Charakterisierung der Hecken aufgenommen. Im gesamten Untersuchungsgebiet wurde eine Biotoptypen- und Nutzungskartierung durchgeführt. Zwischen August und November 2005 wurde die Nahrungsaufnahme von Vögeln an früchtetragenden Sträuchern (Crataegus laevigata, Sorbus aucuparia, Euonymus europaeus) in den Hecken dokumentiert. Die im Gelände erfassten Daten wurden digitalisiert und mittels GIS räumlich und statistisch ausgewertet. Insgesamt wurden 41 Brutreviere von zwölf verschiedenen Vogelarten in den untersuchten Hecken erfasst. Im Mittel wurden 3,6 Brutreviere (n = 9, SD = 1,1) und 3,1 Vogelarten (n = 9, SD = 1,3) pro 100 Meter Hecke festgestellt. Neun Vogelarten (vier Brutvogelarten der Hecken) wurden bei der Nahrungsaufnahme an früchtetragenden Sträuchern erfasst. Als wichtigste Strukturparameter mit Einfluss auf die Abundanz von Brutvögeln in Hecken erwiesen sich Heckenhöhe, Heckenbreite, Anzahl von Überhältern und der Abstand zu Waldbiotopen. Die Diversität von Brutvögeln in Hecken wurde maßgeblich durch die Parameter Heckenhöhe und Heckenlänge beeinflusst. Aus den Ergebnissen werden Pflegeempfehlungen zur Optimierung der Hecken am Rodderberg als Lebensraum für Brutvögel abgeleitet.

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33

Lee, Min Ho, and Hyo Chul Myung. "Hecke Operators on Jacobi-like Forms." Canadian Mathematical Bulletin 44, no. 3 (September 1, 2001): 282–91. http://dx.doi.org/10.4153/cmb-2001-028-6.

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AbstractJacobi-like forms for a discrete subgroup are formal power series with coefficients in the space of functions on the Poincaré upper half plane satisfying a certain functional equation, and they correspond to sequences of certain modular forms. We introduce Hecke operators acting on the space of Jacobi-like forms and obtain an explicit formula for such an action in terms of modular forms. We also prove that those Hecke operator actions on Jacobi-like forms are compatible with the usual Hecke operator actions on modular forms.

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34

Hohnen, Winfried. "Period polynomials and congruences for Hecke algebras." Proceedings of the Edinburgh Mathematical Society 42, no. 2 (June 1999): 217–24. http://dx.doi.org/10.1017/s0013091500020204.

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Using the Eichler-Shimura isomorphism and the action of the Hecke operator T2 on period polynomials, we shall give a simple and new proof of the following result (implicitly contained in the literature): let f be a normalized Hecke eigenform of weight k with respect to the full modular group with eigenvalues λp under the usual Hecke operators Tp (p a prime). Let Kf be the field generated over Q by the λp for all p. Let p be a prime of Kf lying above 5. Then

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35

Wang, Kun, Shanshan Qi, Haitao Ma, and Zhujun Zheng. "Stability for Representations of Hecke Algebras of Type A." Mathematics 10, no. 1 (December 22, 2021): 32. http://dx.doi.org/10.3390/math10010032.

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In this paper, we introduce the concept of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence of the representations of Hecke algebras is representation stable.

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36

LANDSTAD, MAGNUS B., and NADIA S. LARSEN. "GENERALIZED HECKE ALGEBRAS AND C*-COMPLETIONS." International Journal of Mathematics 20, no. 01 (January 2009): 45–76. http://dx.doi.org/10.1142/s0129167x09005169.

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For a Hecke pair (G, H) and a finite-dimensional representation σ of H on Vσ with finite range, we consider a generalized Hecke algebra [Formula: see text], which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (non-full) projection [Formula: see text] such that [Formula: see text] is isomorphic to [Formula: see text]. We study the structure and properties of C*-completions of the generalized Hecke algebra arising from this corner realisation, and via Morita–Fell–Rieffel equivalence, we identify, in some cases explicitly, the resulting proper ideals of [Formula: see text]. By letting σ vary, we can compare these ideals. The main focus is on the case with dim σ = 1 and applications include ax + b-groups and the Heisenberg group.

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37

Demir, Bilal, Özden Koruoğlu, and Recep Sahin. "On Normal Subgroups of Generalized Hecke Groups." Analele Universitatii "Ovidius" Constanta - Seria Matematica 24, no. 2 (June 1, 2016): 169–84. http://dx.doi.org/10.1515/auom-2016-0035.

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Abstract We consider the generalized Hecke groups Hp,q generated by X(z) = -(z -λp)-1, Y (z) = -(z +λq)-1 with and where 2 ≤ p ≤ q < ∞, p+q > 4. In this work we study the structure of genus 0 normal subgroups of generalized Hecke groups. We construct an interesting genus 0 subgroup called even subgroup, denoted by . We state the relation between commutator subgroup H′p,q of Hp,q defined in [1] and the even subgroup. Then we extend this result to extended generalized Hecke groups H̅p,q.

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38

Savage, Alistair. "Affine Wreath Product Algebras." International Mathematics Research Notices 2020, no. 10 (May 24, 2018): 2977–3041. http://dx.doi.org/10.1093/imrn/rny092.

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Abstract We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. In particular, special cases include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate affine Hecke–Clifford algebras), and wreath Hecke algebras. In some cases, specializing the results of the current paper recovers known results, but with unified and simplified proofs. In other cases, we obtain new results, including proofs of two open conjectures of Kleshchev and Muth.

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Rostam, Salim. "Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras." Advances in Mathematics 311 (April 2017): 662–729. http://dx.doi.org/10.1016/j.aim.2017.03.004.

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40

Hida, Haruzo. "Transcendence of Hecke operators in the big Hecke algebra." Duke Mathematical Journal 163, no. 9 (June 2014): 1655–81. http://dx.doi.org/10.1215/00127094-2690478.

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41

UENO, Kimio, and Youichi SHIBUKAWA. "CHARACTER TABLE OF HECKE ALGEBRA OF TYPE AN-1 AND REPRESENTATIONS OF THE QUANTUM GROUP Uq(gln+1)." International Journal of Modern Physics A 07, supp01b (April 1992): 977–84. http://dx.doi.org/10.1142/s0217751x92004130.

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A q-analogue of the Frobenius formula is proved by means of the quantum groups Uq(gln+1), Aq(GLn+1) and Iwahori's Hecke algebra of type AN-1, and then, the character table of this Hecke algebra is investigated.

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42

Baker, Andrew. "Hecke Operations and the Adams E2-Term Based on Elliptic Cohomology." Canadian Mathematical Bulletin 42, no. 2 (June 1, 1999): 129–38. http://dx.doi.org/10.4153/cmb-1999-015-2.

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AbstractHecke operators are used to investigate part of the E2-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of Ext1 which combines use of classical Hecke operators and p-adic Hecke operators due to Serre.

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43

KHAN, RIZWANUR. "The fifth moment of Hecke L-functions in the weight aspect." Mathematical Proceedings of the Cambridge Philosophical Society 168, no. 3 (January 14, 2019): 543–66. http://dx.doi.org/10.1017/s0305004118000944.

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AbstractWe prove an upper bound for the fifth moment of Hecke L-functions associated to holomorphic Hecke cusp forms of full level and weight k in a dyadic interval K ≤ k ≤2K, as K → ∞. The bound is sharp on Selberg’s eigenvalue conjecture.

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44

Bowman, Chris, Anton Cox, Amit Hazi, and Dimitris Michailidis. "Path combinatorics and light leaves for quiver Hecke algebras." Mathematische Zeitschrift 300, no. 3 (September 29, 2021): 2167–203. http://dx.doi.org/10.1007/s00209-021-02829-0.

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AbstractWe recast the classical notion of “standard tableaux" in an alcove-geometric setting and extend these classical ideas to all “reduced paths" in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias–Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the “Bott–Samelson truncation" of the quiver Hecke algebra.

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45

Murin, Max, and Seth Shelley-Abrahamson. "Parameters for generalized Hecke algebras in type B." Journal of Algebra and Its Applications 18, no. 09 (July 17, 2019): 1950173. http://dx.doi.org/10.1142/s0219498819501731.

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The irreducible representations of full support in the rational Cherednik category [Formula: see text] attached to a Coxeter group [Formula: see text] are in bijection with the irreducible representations of an associated Iwahori–Hecke algebra. Recent work has shown that the irreducible representations in [Formula: see text] of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper, we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup [Formula: see text] in [Formula: see text] for [Formula: see text] and [Formula: see text].

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46

CARNEY, ALEXANDER, ANASTASSIA ETROPOLSKI, and SARAH PITMAN. "POWERS OF THE ETA-FUNCTION AND HECKE OPERATORS." International Journal of Number Theory 08, no. 03 (April 7, 2012): 599–611. http://dx.doi.org/10.1142/s1793042112500339.

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Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono in [K. Ono, The partition function and Hecke operators, Adv. Math.228 (2011) 527–534], here we obtain closed formulas for the Hecke images of all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. In addition, congruences for a large class of powers of Ramanujan's Delta-function are obtained in a corollary. We further exhibit a fast calculation for many large values of vector partition functions.

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47

Ohara, Kazuma. "Hecke algebras for tame supercuspidal types." American Journal of Mathematics 146, no. 1 (February 2024): 277–93. http://dx.doi.org/10.1353/ajm.2024.a917543.

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abstract: Let $F$ be a non-archimedean local field of residue characteristic $p\neq 2$. Let $G$ be a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. In~2001, Yu constructed types which are called {\it tame supercuspidal types} and conjectured that Hecke algebras associated with these types are isomorphic to Hecke algebras associated with depth-zero types of some twisted Levi subgroups of $G$. In this paper, we prove this conjecture. We also prove that the Hecke algebra associated with a {\it regular supercuspidal type} is isomorphic to the group algebra of a certain abelian group.

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GOLDMAKHER, LEO, and BENOÎT LOUVEL. "A quadratic large sieve inequality over number fields." Mathematical Proceedings of the Cambridge Philosophical Society 154, no. 2 (October 3, 2012): 193–212. http://dx.doi.org/10.1017/s0305004112000370.

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AbstractWe formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.

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Gallenkämper, Jonas, Bernhard Heim, and Aloys Krieg. "The Maaß space and Hecke operators." International Journal of Mathematics 27, no. 05 (May 2016): 1650039. http://dx.doi.org/10.1142/s0129167x16500397.

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We give a new proof of the fact that the Maaß space is invariant under all Hecke operators. It is based on the characterization of the Maaß space by a symmetry relation and certain commutation relations of the Hecke algebra for the Jacobi group.

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SHEMANSKE, T., S. TRENEER, and L. WALLING. "CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS." International Journal of Number Theory 06, no. 05 (August 2010): 1117–37. http://dx.doi.org/10.1142/s1793042110003411.

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It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

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