# Lectures on differential Galois theory by Andy R. Magid

By Andy R. Magid

Differential Galois thought reviews suggestions of differential equations over a differential base box. In a lot an identical method that standard Galois concept is the speculation of box extensions generated by means of options of (one variable) polynomial equations, differential Galois thought seems to be on the nature of the differential box extension generated via the ideas of differential equations. an extra characteristic is that the corresponding differential Galois teams (of automorphisms of the extension solving the bottom and commuting with the derivation) are algebraic teams. This publication offers with the differential Galois conception of linear homogeneous differential equations, whose differential Galois teams are algebraic matrix teams. as well as offering a handy route to Galois conception, this strategy additionally ends up in the confident resolution of the inverse challenge of differential Galois concept for varied periods of algebraic teams. delivering a self-contained improvement and plenty of particular examples, this publication offers a different method of differential Galois concept and is acceptable as a textbook on the complicated graduate point

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2 Property (T) and Fell’s topology 35 and a + b = 1. We then have ϕ = aψ + bψ . We claim that either b = 0 or ψ = 0. Indeed, assume by contradiction that b = 0 and ψ (e) = 0. Then ψ (e) = 1, since ϕ(e) = 1 and ψ(e) ≤ 1, ψ (e) ≤ 1. 6 that limi ψi = ψ uniformly on compact subsets of G. 8 shows that ψ is a sum of functions of positive type associated to π0 . 1) and this is a contradiction. Therefore, we have limi ai ψi = ϕ and limi ai = 1. Thus, limi ψi = ϕ in the weak* topology; by Raikov’s Theorem, this holds also uniformly on compact subsets of G.

12). 12 again. Since π0 ⊗ π 0 = π1 ⊕ · · · ⊕ πn , it follows that σi is unitarily equivalent to one of the πk ’s. This is a contradiction to the choice of (σi )i∈I . P. 1]. 3 Compact generation and other consequences The ﬁrst spectacular application of Property (T) is the following result, due to Kazhdan. 1 Let G be a locally compact group with Property (T). Then G is compactly generated. In particular, a discrete group with Property (T) is ﬁnitely generated. 3 Compact generation and other consequences 37 Proof Let C be the set of all open and compactly generated subgroups of G.

Indeed, SLn (Q) is not ﬁnitely generated, since every ﬁnite subset {x1 , . . , xm } of SLn (Q) is contained in SLn (Z[1/N ]), where N is a common multiple of the denominators of the matrix coefﬁcients of x1 , . . , xm . 1. 4 Property (T) for SLn (K), n ≥ 3 Let K be a local ﬁeld . 4, for more details). 6). 4. Some general facts We collect the common ingredients used in the proofs of Property (T) for SLn (K) and Sp2n (K). 1 Let G be a topological group, and let (π , H) be a unitary representation of G with 1G ≺ π.