Abstract
This chapter introduces the notion which is the main interest in this book and is responsible for the title. Locally mixed symmetric spaces are a very natural construction which enrich the notion of locally symmetric spaces, studied in Chap. 2. While for locally symmetric spaces there is a pair of data entering, \((G_{\mathbb Q}, {\varGamma })\), where \(G_{\mathbb Q}\) is a semisimple \({\mathbb Q}\)-group such that \(X=G_{\mathbb R}/K\) is a symmetric space of non-compact type for a maximal compact subgroup \(K\subset G_{\mathbb R}\) and \( {\varGamma }\subset G_{\mathbb Q}\) is an arithmetic group, there is now a triple defining the situation: \((G_{\mathbb Q}, {\varGamma },{\boldsymbol{\rho }})\), where \({\boldsymbol{\rho }}:G_{\mathbb Q}\longrightarrow GL(V)\) is a faithful rational representation (not necessarily defined over \({\mathbb Q}\)).
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Notes
- 1.
(Is the imaginary part of a hermitian form of which the real part is symmetric).
- 2.
Over a field any symplectic from is equivalent to the standard one; explicitly \(Q={0\ \ \ \ \varDelta _\delta \atopwithdelims ()-\varDelta _\delta \ 0},\ j_Q={1 \ \ 0 \atopwithdelims ()0\ \varDelta _\delta ^{-1}} \Rightarrow {^t\!j}_Q Q j_Q=\mathbf {J}\), so this is just a notational distinction. In particular the discrete group \(Sp_Q({\mathbb Z})\) is commensurable with \(Sp_{2n}({\mathbb Z})\).
- 3.
A point of potential confusion concerning these groups is the following: since D has dimension \(d^2\) over K, it is not the case that \(SU(D^m,\varPhi )\otimes {\mathbb R}\) and the group \(SU(D^m\otimes {\mathbb R}, \varPhi \otimes {\mathbb R})\) are the same, since the latter is an SU on a \(m\cdot d^2\)-dimensional vector space (over \({\mathbb C}\) when \(d\not =2\) or \({\mathbb R}\) when \(d=2\)). Rather letting \(L\subset D\) denote a splitting field for D, one has \( SU(D^m,\varPhi )\otimes {\mathbb R}\cong SU(L^m\otimes {\mathbb R}, \varPhi _L\otimes {\mathbb R})\), which has irreducible factors \(SU(V^\sigma ,\varPhi ^\sigma )\) with \(V^\sigma \cong {\mathbb C}^{m\cdot d}\) for \(d\not =2\) (resp. \(V^\sigma \cong {\mathbb R}^{2m}\) for \(d=2\)).
- 4.
When G is anisotropic, \(S=\{0\},\ \varDelta =\varTheta _\alpha \) in what follows and \(P^{\varTheta _\alpha }=\) is a minimal parabolic of \(G'\); then G is the anisotropic component of \(Z(S')\) in (6.76).
- 5.
For simplicity of notation the representation \({\boldsymbol{\rho }}\) is suppressed.
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Hunt, B. (2021). Locally Mixed Symmetric Spaces. In: Locally Mixed Symmetric Spaces. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-69804-1_3
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DOI: https://doi.org/10.1007/978-3-030-69804-1_3
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