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Figure 1.1 The subset in .NET Drawer ANSI/AIM Code 39 in .NET Figure 1.1 The subset




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Figure 1.1 The subset generate, create code39 none with .net projects Postnet of K 2 1.4 Property (T) for SLn (K), n 3 x y Indeed, for , x y x + n y y gn and x + n y n y x. (. n . 1). y. , . x + n y x. + . n y (. n . + 1). y. . Since 1 1 < n . 1 . m . + 1 the sets for n > m,. are pairwise disjoint. Hence,. m(K 2 \ {0}) 1. for all n N. As m( =. i ) = m(gi ) = m( ), it f ollows that m( ) = 0. If = x y K 2 \ {0} : . x. y. ,. 0 1. then m( ) = m( .net vs 2010 Code39 ) = 0. Since = K 2 \ {0}, we have m(K 2 \ {0}) = 0.

Therefore, m is the Dirac measure at 0. Corollary 1.4.

13 The pair (SL2 (K) local eld K. K 2 , K 2 ) has Property (T), for every. Remark 1.4.14 The semidirect product SL2 (K) K 2 does not have Property (T).

Indeed, SL2 (K) is a quotient of SL2 (K) K 2 and does not have Property (T); see Example 1.3.7 for the case K = R and Example 1.

7.4 for the other cases. We are now ready to show that SLn (K) has Property (T) for n 3.

Theorem 1.4.15 Let K be a local eld.

The group SLn (K) has Property (T) for any integer n 3. Proof Let ( , H) be a unitary representation of SLn (K) almost having invariant vectors. Let G SL2 (K) K 2 and N K 2 be the subgroups of SLn (K) = =.

Property (T). introduced abo ve. By the previous corollary, the pair (G, N ) has Property (T). Hence, there exists a non-zero N -invariant vector H.

By Lemma 1.4.9, is invariant under the following copy of SL2 (K) 0 0 0 1 0 0 0 0 0 0 0 In 3 .

inside SLn (K) 3 of 9 for .NET . It follows from Proposition 1.

4.11 that is invariant under the whole group SLn (K). Other examples of groups with Property (T) are provided by the following corollary and by Exercises 1.

8.6 1.8.

10. Corollary 1.4.

16 Let K be a local eld. The semidirect product G = SLn (K) K n has Property (T) for n 3. Proof Let ( , H) be a unitary representation of G almost having invariant vectors.

Since SLn (K) has Property (T), there exists a non-zero vector H which is SLn (K)-invariant. For every x K n , there exists a sequence Ai SLn (K) with limi Ai x = 0. It follows from Mautner s Lemma 1.

4.8 that is invariant under K n . Hence, is invariant under SLn (K) K n .

. 1.5 Property (T) for Sp2n (K), n 2 In this sectio VS .NET USS Code 39 n, we prove that the symplectic group Sp2n (K) has Property (T) for n 2. The strategy of the proof is similar to that for SLn (K): we show that an appropriate subgroup of Sp2n (K) gives rise to a pair which has Property (T).

Recall that Sp2n (K) is the closed subgroup of GL2n (K) consisting of all matrices g with t gJg = J , where t g is the transpose of g, J = 0 In In 0 ,. and In is the n n identity matrix. Observe that Sp2 (K) = SL2 (K); see Exercise 1.8.

1. Writing matrices in GL2n (K) as blocks g= A C B D. 1.5 Property (T) for Sp2n (K), n 2 of (n n) matrices, we have: g Sp2n (K) if and only if AC t CA = t BD t DB = 0 AD t CB = I . Let S 2 (K 2 Code 3/9 for .NET ) be the vector space of all symmetric bilinear forms on K 2 . The group GL2 (K) acts on S 2 (K 2 ) by g , where.

(x, y) = (t gx, t gy). for g GL2 (K 3 of 9 for .NET ) and x, y K 2 . Each S 2 (K 2 ) is of the form (x, y) = X x, y for a unique symmetric (2 2) matrix X with coef cients in K, where x, y = x1 y1 + x2 y2 is the standard symmetric bilinear form on K 2 .

The matrix corresponding to g is then gX t g. Consider the subgroups G2 = A 0 0 0 0 In 2 0 0 0 In 2 0 0 0 0 t A 1 0 B 0 I2 0 0 0 0 In 2 0 0 0 In 2 : A SL2 (K) SL2 (K) = .
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