ON THE CENTER OF A HECKE ALGEBRA

J. L. ALPERIN

In representation theory, centers play special roles and we shall be looking at the case of Hecke algebras. Let G be a finite group, X a transitive G-set and H_{ℤ} the corresponding Hecke algebra, namely, the endomorphism ring of the ℤG-module ℤX. The center of the group algebra ℤG induces endomorphisms of ℤX which commute with all endomorphisms so has an image Z_{0} in the center Z of H_{ℤ}. Our purpose is to establish the following result:

Theorem. The quotient Z/Z_{0} is finite of order equal to the product of the elementary divisors of the reduced class-coset table of H_{ℤ}.

The class-coset table is the matrix whose rows are indexed by the conjugacy classes of G and whose columns are indexed by the double cosets of H in G, where H is the stablizer of the point x of X, and where an entry is given by the number of elements in a conjugacy class belonging to a double coset. (This table is already important in the complex representation theory of H_{ℤ} because the character table of the complex Hecke algebra is the matrix obtained from the product of 1/|H| with the character table of G times the class-coset table, after deletion of rows of zeros.) We let the reduced table be obtained by dividing each entry of this table by the number of cosets in the corresponding double coset. Since the number of elements in one conjugacy class lying in a coset is constant over the cosets in one double coset, the entries of the reduced table are also natural numbers. (If the reduced table is used in a matrix product, as above, one gets the spherical functions of the Hecke algebra instead of the characters.)

The two rings, Z and Z_{0}, are subgroups of the additive group of H_{ℤ} and we assert they have the same rank. This is equivalent to repeating the constructions of the Hecke algera and these rings over the rational numbers and seeing that there the two rings agree. This then follows from the same property over the complex numbers since we are dealing with semisimple algebras. The standard structure results in that case [1] yield equality. Hence, the quotient Z/Z_{0} is finite. Let W/Z_{0} be the torsion subgroup of the abelian group H_{ℤ}/Z_{0}. If w is an element of W and h is in H_{ℤ} then the additive commuator [w, h] will be of finite order in the additive group H_{ℤ} so will be zero and W is contained in Z. Hence, W = Z.

It remains now to connect the torsion subgroup W/Z_{0} with the class-coset table. Let g be an element of G, K the sum of the elements conjugate with g. Any endomorphism of the ℤG-module ℤX sends x to a linear combination of sums of the orbits of H on X. Let us calculate Kx in terms of such orbit sums. Let Y be one such orbit sum, so that Y is the sum of all elements in Hax, for some a in G. The coefficient of Y in the expression of Kx in terms of orbit sums is calculated as follows. If k is in K then kx lies in Hax if, and only if, k is in HaH so the coefficient of Y is the cardinality of K ∩HaH divided by the cardinality of Y, i.e. the number of cosets of H in the double coset HaH, which is an entry in the reduced class-coset table in the row for K. Since the action of K on ℤX is determined by the image of x, the proof is complete.

Another approach is possible using the description of the Hecke algebra inside of the complex group algebra and the standard basis of double coset sums divided by |H|, by calculating eKe, where e is the idempotent \frac{1}{|H|} ∑_{h∈H} h.

References

[1] Curtis, Charles; Reiner, Irving; Methods of representation theory. Vol I. With applications to finite groups and orders. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1981, pp. 279-291.