By Kalantari I., Welch L.

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**Example text**

ShB is a projective limit of varieties ShB,W , where W is an open compact subgroup of the ﬁnite adele group of the reductive Q-group G = GB = ResF/Q (B∗) associated to the multiplicative group of B. Each ShB,W is deﬁned over the canonically deﬁned number ﬁeld F , named by Shimura the reﬂex ﬁeld; the deﬁnition is recalled below in Section 3. We say that WMC holds for ShB if it holds for each smooth variety ShB,W . 2. The Shimura variety is not proper exactly when B = M2 (F ), in which case the connected components of the ShB,W are the classical Hilbert modular varieties.

L7] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series 1 8 , Amer. Math. Soc. 2003. [Mi] P. Mischenko, Invariant tempered distributions on the reductive p-adic group GLn (Fp ), C. R. Math. Rep. Acad. Sci. Canada 4 (1982),123–127. [M1] L. Morris, Tamely ramiﬁed intertwining algebras, Invent. Math. 1 1 4 (1993), 1–54. [M2] L. Morris, Level zero G-types, Compositio Math. 1 1 8 (1999), 135–157. [N] J. Neukirch, Algebraic number theory, Springer-Verlag, 1999. J. Plymen, Reduced C ∗ -algebra of the p-adic group GL(n) II, J.

Mi] P. Mischenko, Invariant tempered distributions on the reductive p-adic group GLn (Fp ), C. R. Math. Rep. Acad. Sci. Canada 4 (1982),123–127. [M1] L. Morris, Tamely ramiﬁed intertwining algebras, Invent. Math. 1 1 4 (1993), 1–54. [M2] L. Morris, Level zero G-types, Compositio Math. 1 1 8 (1999), 135–157. [N] J. Neukirch, Algebraic number theory, Springer-Verlag, 1999. J. Plymen, Reduced C ∗ -algebra of the p-adic group GL(n) II, J. Functional Analysis 1 9 6 (2002) 119–134. [Re] M. Reeder, Isogenies of Hecke algebras and a Langlands correspondence for ramiﬁed principal series representations, Represent.