\(A_{\mathfrak{q}}\)-components of geometric classes in compact Hermitian locally symmetric spaces. (English) Zbl 1527.22020

Mondal, Arghya

Kyoto J. Math. 63, No. 1, 51-66 (2023).

Summary: Let \(\Gamma \setminus G/ K\) be a compact Hermitian locally symmetric space, where \(G\) is simple. We study the components of a de Rham cohomology class of \(\Gamma \setminus G/ K\) with respect to the Matsushima decomposition, where the class is obtained by taking the Poincaré dual of a totally geodesic complex analytic submanifold. Using an improved version of the vanishing result of Kobayashi and Oda, we specify the existence of certain components of such cohomology classes when \(G=\mathrm{SU}(p,q)\).

MSC:

22E40	Discrete subgroups of Lie groups
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols

Keywords:

automorphic representations; geometric cycles; vanishing theorem

References:

[1]	A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111-122. · Zbl 0116.38603 · doi:10.1016/0040-9383(63)90026-0
[2]	A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. Math. Stud. 94, Princeton Univ. Press, Princeton, 1980. · Zbl 0443.22010
[3]	D. DeGeorge and N. Wallach, Limit formulas for multiplicities in \[{L^2}(\Gamma \setminus G)\], Ann. of Math. (2) 107 (1978), no. 1, 133-150. · Zbl 0397.22007 · doi:10.2307/1971140
[4]	I. M. Gelfand and I. I. Pyatetskii-Shapiro, Theory of representations and theory of automorphic functions, Amer. Math. Soc. Transl. (2) 26 (1963), 173-200. · Zbl 0138.31901
[5]	S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, corrected reprint, Grad. Stud. Math 34, Amer. Math. Soc., Providence, 2001. · Zbl 0993.53002 · doi:10.1090/gsm/034
[6]	T. Kobayashi, Discrete decomposability of the restriction of \[{A_{\mathfrak{q}}}(\lambda )\]with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), no. 2, 181-202. · Zbl 0826.22015 · doi:10.1007/BF01232239
[7]	T. Kobayashi, Discrete decomposability of the restriction of \[{A_{\mathfrak{q}}}(\lambda )\]with respect to reductive subgroups III, Invent. math. 131 (1997), no. 2, 229-256. · Zbl 0907.22016 · doi:10.1007/s002220050203
[8]	T. Kobayashi, A vanishing theorem for modular symbols on locally symmetric spaces II, preprint, 2009.
[9]	T. Kobayashi and T. Oda, A vanishing theorem for modular symbols on locally symmetric spaces, Comment. Math. Helv. 73 (1998), 45-70. · Zbl 0933.11027 · doi:10.1007/s000140050045
[10]	T. Kobayashi and Y. Oshima, Classification of discretely decomposable \[{A_{\mathfrak{q}}}(\lambda )\]with respect to reductive symmetric pairs, Adv. Math. 231 (2012), nos. 3-4, 2013-2047. · Zbl 1255.22010 · doi:10.1016/j.aim.2012.07.006
[11]	S. S. Kudla and J. J. Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Inst. Hautes Études Sci. Publ. Math. 1990, no. 71, 121-172. · Zbl 0722.11026
[12]	J. J. Millson and M. S. Raghunathan, Geometric construction of cohomology for arithmetic groups I, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 2, 103-123. · Zbl 0524.22012 · doi:10.1007/BF02837282
[13]	A. Mondal and P. Sankaran, Geometric cycles in compact locally Hermitian symmetric spaces and automorphic representations, Transform. Groups. 24 (2019), no. 3, 913-948. · Zbl 1433.22008 · doi:10.1007/s00031-019-09535-6
[14]	C. C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966), 154-178. · Zbl 0148.37902 · doi:10.2307/2373052
[15]	M. S. Raghunathan, Arithmetic lattices in semisimple groups, Proc. Indian Acad. Sci. Math. Sci. 91 (1982), no. 2, 133-138. · Zbl 0491.22006 · doi:10.1007/BF02967980
[16]	T. Oda and M. Tsuzuki, The secondary spherical functions and Green currents associated with certain symmetric pairs, Pure Appl. Math. Q. 5 (2009), no. 3, 977-1028. · Zbl 1257.11049 · doi:10.4310/PAMQ.2009.v5.n3.a4
[17]	J. Rohlfs and J. Schwermer, Intersection numbers of special cycles, J. Amer. Math. Soc. 6 (1993), 755-778. · Zbl 0811.11039 · doi:10.2307/2152782
[18]	M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 5, 3rd ed., Publish or Perish, Houston, 1999.
[19]	Y. L. Tong and S. P. Wang, Geometric realization of discrete series for semisimple symmetric spaces, Invent. Math. 96 (1989), no. 2, 425-458. · Zbl 0685.22008 · doi:10.1007/BF01393969
[20]	M. Tsuzuki, Limit formulas of period integrals for a certain symmetric pair, J. Funct. Anal. 255 (2008), no. 5, 1139-1190. · Zbl 1158.43009 · doi:10.1016/j.jfa.2008.05.017
[21]	D. A. Vogan and G. J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51-90. · Zbl 0692.22008