By O. T. O'Meara

This quantity, the sequel to the author's Lectures on Linear teams, is the definitive paintings at the isomorphism conception of symplectic teams over fundamental domain names. lately came across geometric tools that are either conceptually easy and strong of their generality are utilized to the symplectic teams for the 1st time. there's a whole description of the isomorphisms of the symplectic teams and their congruence subgroups over necessary domain names. Illustrative is the theory $\mathrm{PSp}_n(\mathfrak o)\cong\mathrm{PSp}_{n_1}(\mathfrak o_1)\Leftrightarrow n=n_1$ and $\mathfrak o\cong\mathfrak o_1$ for dimensions $\geq 4$. the recent geometric technique utilized in the ebook is instrumental in extending the idea from subgroups of $\mathrm{PSp})n(n\geq6)$ the place it used to be identified to subgroups of $\mathrm{P}\Gamma\mathrm{Sp}_n(n\geq4)$ the place it really is new. There are wide investigations and a number of other new effects at the unprecedented habit of $\mathrm{P}\Gamma\mathrm{Sp}_4$ in attribute 2. the writer starts off primarily from scratch (even the classical simplicity theorems for $\mathrm{PSp}_n(F)$ are proved) and the reader want be accustomed to not more than a primary direction in algebra.

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**Example text**

16. Let char F = 2. i4) where A' is invertible and diagonal if (J with A = in SPn (V), there is a (40) not hyperbolic, and A' has the form uL 01 1 + 1 I 0 If (J is hyperbolic. 13. Now R = p* so so it follows easily that P contains a totally degenerate subspace of dimension n that contains R. Accordingly, by 1. 14, there is a symplectic base PROOF. R = rad P, i GENERATION THEOREMS 29 for V in which R = Fx I + . . + C Fx I FXr + . . + FXr + . . + FXn / 2 CP. 17 we know that a-(~) inID for some ~ n X ~ n symmetric matrix D.

0; hence k 2x = mkx. and we have (3). So P = 1 and O. 54 T. 2. If k is a hyperbolic transformation in rSp,,( V), then k is in Sp,,( V) only if k is an involution. PROOF. 1. D. 3. If a hyperbolic transformation k in rSp,,( V) stabilizes a line in V, then k is in PSp,,( V). PROOF. 4. D. 4. Let k be a hyperbolic transformation in rSp,,(V), and hence in GSp,,(V), . 2 such that m, E F - F . A) where A is the ~ n X ~ n matrix PROOF. 4 we know that k f/:. 3 we know that k can stabilize no line in V.

So 1= q(x,p) = ma-1q(ax, ap) = ma-1q(ax,p) = ma-I. So a is in SPn( V). (2) Take PI' P2 in P with q(pI' P2) = I. D. 7. Every transvection in fSPn( V) is already in SPn( V). Every projective 53 SYMPLECTIC COLLINEAR TRANSFORMATIONS transvection in prSPn( V) is already in PSPn( V), and its representative transvection is in SPn( V). PROOF. 3. So assume n > 4. If a is a transvection in rSPn( V), then a E GSPn( V), and q(P. P) =1= 0 since dim P > n - 1 > ~ n. 6. A projective transvection in prsPn (V) has the form T = k with T a transvection in SL n( V) and k an element of rSPn( V).