I said that the "K-datum" discussed this week for a K-type is essentially (\lambda + \theta\lambda)/2, with \lambda from an atlas parameter. This code makes a function lambdaNotes which takes the mth parameter from a list of parameters and spits out the vector above. One should think of the vector as living in the dual of a Cartan t for k, sitting inside the dual of a fundamental Cartan h. For GL(2n), a vector in the dual of h has 2n coordinates v_1,,...,v_2n. It belongs to t^* if and only if the last n are the negatives of the first n; in this case, the coordinates used in OneNotes are (2v_1,...,2v_n). So when looking at the output of lambdaNotes for GL(2n), you should just multiply the first n coordinates by 2, and discard the rest. atlas> set G=GL(8,R) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> set B=block_of(G.trivial) Variable B: [Param] (overriding previous instance, which had type [Param]) atlas> #B Value: 2835 atlas> set p=B[1400]*0 {*0 kills nu} Variable p: Param atlas> set q =first_param (finalize(p)) Variable q: Param atlas> set lambdaNotes=(q.lambda + q.x.involution*q.lambda)/2 Variable lambdaNotes: ratvec atlas> p Value: non-dominant parameter(x=508,lambda=[7,7,3,3,-1,-3,-5,-7]/2,nu=[0,0,0,0,0,0,0,0]/1) atlas> B[1400] Value: final parameter(x=508,lambda=[7,7,3,3,-1,-3,-5,-7]/2,nu=[5,5,5,1,-4,-4,-4,-4]/2) atlas> p Value: non-dominant parameter(x=508,lambda=[7,7,3,3,-1,-3,-5,-7]/2,nu=[0,0,0,0,0,0,0,0]/1) atlas> q Value: final parameter(x=0,lambda=[3,3,1,1,1,-1,-1,-3]/2,nu=[0,0,0,0,0,0,0,0]/1) atlas> lambdaNotes Value: [ 3, 2, 1, 0, 0, -1, -2, -3 ]/2 So this one is [3,2,1,0]: n_3 = n_2 = n_1 = 1, and n_0 = 1. atlas> set lambdaNotes(int m) = let q=first_param(finalize(B[m]*0)) in (q.lambda + q.x.involution*q.lambda)/2 Defined lambdaNotes: (int->ratvec) This is now a function that takes the mth element in a list of parameters and extracts the lambda from the notes. First check that it gives the same answer we got by hand in the example: atlas> lambdaNotes(1400) Value: [ 3, 2, 1, 0, 0, -1, -2, -3 ]/2 atlas> for i:10 from 1500 do lambdaNotes(i) od Value: [[ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2, [ 5, 0, 0, 0, 0, 0, 0, -5 ]/2] atlas> {GOT: [5,0,0,0] in each of these ten cases. This is an example of a common phenomenon in big blocks: many of the parameters in them have the _same_ lowest K-type Now print it a bit better:} atlas> void: for i:100 from 1600 do prints(i, " ",lambdaNotes(i)) od 1600 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1601 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1602 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1603 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1604 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1605 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1606 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1607 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1608 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1609 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1610 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1611 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1612 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1613 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1614 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1615 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1616 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1617 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1618 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1619 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1620 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1621 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1622 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1623 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1624 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1625 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1626 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1627 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1628 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1629 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1630 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1631 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1632 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1633 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1634 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1635 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1636 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1637 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1638 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1639 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1640 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1641 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1642 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1643 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1644 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1645 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1646 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1647 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1648 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1649 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1650 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1651 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1652 [ 3, 2, 0, 0, 0, 0, -2, -3 ]/2 1653 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1654 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1655 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1656 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1657 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1658 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1659 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1660 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1661 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1662 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1663 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1664 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1665 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1666 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1667 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1668 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1669 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1670 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1671 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1672 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1673 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1674 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1675 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1676 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1677 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1678 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1679 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1680 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1681 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1682 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1683 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1684 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1685 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1686 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1687 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1688 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1689 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1690 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1691 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1692 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1693 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1694 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1695 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1696 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1697 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1698 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 1699 [ 4, 1, 0, 0, 0, 0, -1, -4 ]/2 atlas> void: for i:100 from 2400 do prints(i, " ",lambdaNotes(i)) od 2400 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2401 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2402 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2403 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2404 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2405 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2406 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2407 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2408 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2409 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2410 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2411 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2412 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2413 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2414 [ 1, 1, 1, 0, 0, -1, -1, -1 ]/2 2415 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2416 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2417 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2418 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2419 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2420 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2421 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2422 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2423 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2424 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2425 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2426 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2427 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2428 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2429 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2430 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2431 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2432 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2433 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2434 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2435 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2436 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2437 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2438 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2439 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2440 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2441 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2442 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2443 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2444 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2445 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2446 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2447 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2448 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2449 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2450 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2451 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2452 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2453 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2454 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2455 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2456 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2457 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2458 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2459 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2460 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2461 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2462 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2463 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2464 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2465 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2466 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2467 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2468 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2469 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2470 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2471 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2472 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2473 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2474 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2475 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2476 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2477 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2478 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2479 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2480 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2481 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2482 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2483 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2484 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2485 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2486 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2487 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2488 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2489 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2490 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2491 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2492 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2493 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2494 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2495 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2496 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2497 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2498 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 2499 [ 1, 0, 0, 0, 0, 0, 0, -1 ]/1 {We can use exactly the same script in U(p,q), just changing the list of parameters:} atlas> set G=U(5,3) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> set B=block_of(G.trivial) Variable B: [Param] (overriding previous instance, which had type [Param]) atlas> #B Value: 1736 atlas> lambdaNotes(1000) Value: [ 2, 2, 0, 0, 0, 0, -2, -2 ]/1 {Since G is equal rank, every coordinate matters. In the notation of the notes, here n=5+3, n1=2, n2=4, n3=2} atlas> void:for i:20 from 1200 do prints(i," ",lambdaNotes(i)) od 1200 [ 5, 1, 0, 0, 0, 0, -1, -5 ]/2 1201 [ 5, 1, 0, 0, 0, 0, -1, -5 ]/2 1202 [ 5, 1, 0, 0, 0, 0, -1, -5 ]/2 1203 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1204 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1205 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1206 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1207 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1208 [ 2, 1, 0, 0, 0, 0, -1, -2 ]/1 1209 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1210 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1211 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1212 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1213 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1214 [ 3, 3, 0, 0, 0, 0, -3, -3 ]/2 1215 [ 5, 1, 0, 0, 0, 0, -1, -5 ]/2 1216 [ 5, 1, 0, 0, 0, 0, -1, -5 ]/2 1217 [ 5, 1, 0, 0, 0, 0, -1, -5 ]/2 1218 [ 5, 1, 0, 0, 0, 0, -1, -5 ]/2 1219 [ 5, 1, 0, 0, 0, 0, -1, -5 ]/2