Q.lilies
“Sprouts” or “shoots” are coming up around my mature lilies. Should I leave them alone or what?

You Lilies have began to spread and these are new shoots from the bulbs.
Yes, leave them to enlarge and enhance your garden.
In the fall you can lift and divide the bulbs if you choose too.
Here are some articles to help you.
https://www.gardeningknowhow.com/ornamental/bulbs/lily/planting-asiatic-lilies.htm
https://www.gardeningknowhow.com/ornamental/bulbs/lily/dividing-lily-plants.htm